#6595053

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1321 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6502249);