#6611523

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1530 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgh,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,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, 6518719);