#6524519

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 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)(3,6)(4,8)(5,7)(17,18,23,37,22,26,21,27)(19,28,36,24,30,43,31,40)(20,32,38,42,34,25,35,47)(29,45,48,41,33,44,46,39);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,33)(21,36)(22,30)(23,31)(25,41)(26,40)(27,43)(28,37)(29,34)(32,39)(35,46)(38,48)(42,44)(45,47);(2,7)(6,8)(10,13)(11,15)(17,20)(18,25)(19,29)(21,35)(22,34)(23,38)(24,39)(26,32)(27,42)(28,44)(30,33)(31,46)(36,48)(37,47)(40,41)(43,45);(9,11,12,15)(10,14,13,16)(20,34)(25,32)(29,33)(35,38)(39,41)(42,47)(44,45)(46,48);(1,4,5,3)(2,8,7,6)(10,13)(14,16)(18,26)(19,30)(20,34)(27,37)(31,36)(35,38)(39,41)(44,45)

Polycyclic presentation:

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

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, 6431715);