#99294

order	512
nilpotency class	3
isoclinism	isoclinic class of #99293
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#383
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=deg,c^2=d,d^2=gh,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,f^b=fh,f^c=fgh

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, 6490);