#93043

order	512
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#327
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#431	C₂²
socle	C₂²	#1294
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×#431 ▷ C₂²×C₄ ▷ C₂² ▷ C₁

268 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=hi,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,e^c=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, 239);