#122361

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
socle	C₂²	#1459
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

326 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=g,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ch,e^b=ef,e^d=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29557);