#93459

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

232 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

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, 655);