#93251

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂×C₄	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#336
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

222 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 56 points:

(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22)(25,26,28,32,29,34,39,46,30,33,40,45,41,47,52,55)(27,35,37,48,38,49,51,56,36,31,42,43,50,44,53,54);(1,2,3,5,4,6,7,8)(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24)(25,27)(26,31,33,35)(28,37)(29,38)(30,36)(32,43,45,48)(34,44,47,49)(39,51)(40,42)(41,50)(46,54,55,56)(52,53)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,b^a=bc,e^a=efi,c^b=ci,d^b=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 447);