#122225

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁴

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	C₂×#323
prime radical	C₂³×D₄	C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

567 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 567 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 64

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

Polycyclic presentation:

2^9|a^2=e,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, 29421);