#124728

order	512
nilpotency class	4
isoclinism	isoclinic class of #124728
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1502
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7614
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

275 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fg,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31924);