#6630697

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629011
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537893);