#149698

order	512
nilpotency class	5
isoclinism	isoclinic class of #13578
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#353
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	#11475
prime radical	#3150	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,d^2=h,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dh,e^b=efhi,f^b=fi,e^d=ehi,f^e=fi

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, 56894);