#153464

order	512
nilpotency class	6
isoclinism	isoclinic class of #153457
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13561
commutator	#127	C₂²×C₄
Frattini, cosocle	#492	C₂³
prime radical	#532	C₂×C₄
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3152 ◁ G
LECS	G ▷ #492 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

297 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,11,7,13,25,39)(3,14,16,37,18,30,44,57)(4,9,21,33,20,35,49,60)(6,10,23,36,26,12,24,38)(8,27,29,45,31,43,56,17)(15,40,41,59,19,46,47,63)(22,34,52,61,50,32,51,58)(28,53,54,48,55,62,64,42);(1,3,7,18)(2,8,13,31)(4,19,20,15)(5,16,25,44)(6,17,26,45)(9,28,35,55)(10,14,12,30)(11,37,39,57)(21,47,49,41)(22,48,50,42)(23,27,24,43)(29,36,56,38)(32,46,34,40)(33,59,60,63)(51,53,52,62)(54,61,64,58);(1,4)(2,9)(3,15)(5,21)(6,22)(7,20)(8,28)(10,34)(11,33)(12,32)(13,35)(14,40)(16,41)(17,42)(18,19)(23,52)(24,51)(25,49)(26,50)(27,53)(29,54)(30,46)(31,55)(36,61)(37,59)(38,58)(39,60)(43,62)(44,47)(45,48)(56,64)(57,63)

Polycyclic presentation:

2^9|a^2=e,b^2=i,e^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ch,e^b=efhi,f^b=fhi,g^b=ghi,d^c=di,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306620, #72306631, #72306632, …
maximal normal subgroup of	#72306620, #72306631, #72306632, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60660);