#93558

order	512
nilpotency class	4
isoclinism	isoclinic class of #93520
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#342
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂×#376	C₂²
prime radical	C₂⁴	#95
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂×#376 ▷ C₂⁴ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 332 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) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=i,e^2=f,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh,g^b=gi,e^c=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, 754);