#6557950

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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

1462 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fi,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465146);