#122265

order	512
nilpotency class	4
isoclinism	isoclinic class of #11796
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
socle	C₂²	C₂×#323
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

464 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9)(4,12,13,7)(5,8)(6,10,14,16)(11,15)(17,18,21,27,22,28,36,45,23,25,37,41,20,26,34,43)(19,29,31,46,30,39,47,44,32,42,35,48,33,24,38,40);(1,3,5,11)(2,6,8,14)(4,10,13,16)(7,9,12,15)(17,19)(18,24)(20,33)(21,35)(22,30)(23,32)(25,39)(26,42)(27,44)(28,29)(31,37)(34,47)(36,38)(40,41)(43,46)(45,48);(18,25)(24,39)(26,28)(27,41)(29,42)(40,44)(43,45)(46,48)

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=i,e^2=g,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,d^b=di,e^b=efi,e^d=eh

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, 29461);