#92881

order	512
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#327
commutator	C₂²×C₈	C₂×C₈
Frattini, cosocle	C₂×#336	C₂²
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂×#336 ▷ C₂³×C₄ ▷ C₂² ▷ C₁

280 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=ei,c^2=fhi,d^2=g,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=defh,f^b=fi,g^b=gi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77);