#122985

order	512
nilpotency class	4
isoclinism	isoclinic class of #122971
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂⁵	D₈
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

442 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 46 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)(4,7)(5,8)(6,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,22,27,30)(20,29,28,24)(23,32,31,26);(1,3,5,11)(2,6,8,13)(4,10,12,16)(7,9,14,15)(18,22)(19,26)(20,28)(23,24)(25,30)(27,32)(29,31);(4,12)(10,16)(19,27)(26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=fh,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30181);