#125036

order	512
nilpotency class	4
isoclinism	isoclinic class of #124849
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	#1467
prime radical	C₂³	#407
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

237 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 128

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

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

Polycyclic presentation:

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

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, 32232);