#153522

order	512
nilpotency class	6
isoclinism	isoclinic class of #153520
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1408
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	#492	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3147 ◁ G
LECS	G ▷ #492 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

286 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 48 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)(3,14,17,38)(4,9)(5,23,25,10)(6,12,28,11)(7,13)(8,16,30,46)(15,41,45,63)(18,43,27,42)(19,48,50,60)(20,51,53,34)(21,36,55,35)(22,37)(24,32)(26,39,57,40)(29,44,33,49)(31,47)(52,58)(54,61,64,62)(56,59);(1,3,7,18)(2,8,13,30)(4,19,22,15)(5,24,26,47)(6,27,28,17)(9,33,37,29)(10,38,40,42)(11,32,12,31)(14,23,43,39)(16,25,46,57)(20,52,54,56)(21,45,55,50)(34,60,62,63)(35,58,36,59)(41,61,48,51)(44,64,49,53);(1,4,7,22)(2,9,13,37)(3,15,18,19)(5,20,26,54)(6,21,28,55)(8,29,30,33)(10,34,40,62)(11,35,12,36)(14,41,43,48)(16,44,46,49)(17,45,27,50)(23,51,39,61)(24,56,47,52)(25,53,57,64)(31,58,32,59)(38,63,42,60)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283179, #72283214, #72283226, …
maximal normal subgroup of	#72283179, #72283214, #72283226, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60718);