#92946

order	512
nilpotency class	4
isoclinism	isoclinic class of #7221
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1279
prime radical	C₂³	D₄⋊C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 181 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fgh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,c^b=ch,d^b=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142);