#122621

order	512
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
socle	C₂²	C₂×#327
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

424 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 78 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)(17,18,20,25,23,27,22,38)(19,28,31,42,32,46,29,45)(21,34,36,26,37,43,33,44)(24,35,39,48,41,47,40,30);(1,3,5,11)(2,6,8,14)(4,10,13,16)(7,9,12,15)(17,19)(18,24)(20,29)(21,35)(22,31)(23,32)(25,39)(26,42)(27,41)(28,34)(30,36)(33,48)(37,47)(38,40)(43,46)(44,45);(17,20,23,22)(18,25,27,38)(19,29,32,31)(21,33,37,36)(24,39,41,40)(26,34,44,43)(28,45,46,42)(30,35,48,47)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

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, 29817);