#93408

order	512
nilpotency class	4
isoclinism	isoclinic class of #7262
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

230 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2)(3,5)(4,7)(6,8)(9,10,13,15,14,11,12,16)(18,23,24,25)(19,27)(20,29)(21,28)(26,32,30,31);(2,5,7,6)(9,11,14,10)(12,15,13,16)(17,18,20,25,21,26,29,32,22,24,19,23,28,30,27,31)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 604);