#93710

order	512
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	#1831	C₂²
socle	C₂²	#1318
prime radical	C₂³	#323
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1831 ◁ G
LECS	G ▷ #1831 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

169 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 169 conjugacy classes of subgroups — too many to draw. Only its 32 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=gh,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cgh,d^b=deghi,e^b=ei,d^c=dgi,e^d=ei

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, 906);