#99714

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	#23421	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 736 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=gh,d^2=g,e^2=hi,f^2=ghi,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6910);