#93712

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	#1842	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 ◁ #1842 ◁ G
LECS	G ▷ #1842 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 166 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)(19,26,27,22)(20,24,30,23)(21,25)(28,31,29,32);(2,6)(8,13)(17,19,21,28)(18,22,25,32)(20,29,30,27)(23,31,24,26)

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=ghi,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cghi,d^b=degh,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, 908);