#92915

order	512
nilpotency class	4
isoclinism	isoclinic class of #7217
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	#1277
prime radical	C₂³	#335
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 160 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=f,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111);