#124619

order	512
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1533
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#376	C₂³
socle	C₂	#7601
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,13,16,37,17,33,25,56)(4,18,20,9)(5,22,23,54)(6,11)(7,12)(8,15,28,43,30,45,29,44)(10,24,34,55)(14,38,40,60,42,36,41,59)(19,49)(21,48,52,32)(26,35)(27,39,57,61,46,62,58,63)(31,50)(47,64)(51,53);(1,3)(2,8)(4,14)(5,15)(6,25)(7,17)(9,27)(10,33)(11,28)(12,30)(13,34)(16,26)(18,46)(19,39)(20,42)(21,40)(22,37)(23,45)(24,43)(29,35)(31,38)(32,58)(36,47)(41,52)(44,55)(48,57)(49,59)(50,63)(51,62)(53,60)(54,56)(61,64);(1,4,6,21,7,20,26,52)(2,9,11,32,12,18,35,48)(3,14,25,40,17,42,16,41)(5,19,24,50,23,51,55,64)(8,27,28,58,30,46,29,57)(10,31,22,53,34,47,54,49)(13,36,56,59,33,38,37,60)(15,39,43,63,45,62,44,61)

Polycyclic presentation:

2^9|a^2=e,c^2=fh,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cg,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,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, 31815);