#6463875

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 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)(17,18,21,25,22,26,35,42)(19,27,30,37,29,41,34,23)(20,24,33,40,32,39,47,48)(28,43,44,38,31,45,46,36);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,31)(21,34)(22,29)(24,38)(25,41)(26,37)(27,42)(28,32)(30,35)(33,44)(36,39)(40,43)(45,48)(46,47);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,40)(26,39)(27,43)(29,31)(30,44)(34,46)(35,47)(37,38)(41,45)(42,48);(9,11,12,16)(10,14,13,15)(19,29)(23,37)(27,41)(28,31)(30,34)(36,38)(43,45)(44,46);(20,32)(24,39)(28,31)(33,47)(36,38)(40,48)(43,45)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371071);