#6463877

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

1446 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)(17,18,22,27,23,28,21,26)(19,29,32,39,31,44,37,24)(20,25,34,41,35,42,36,43)(30,45,47,40,33,48,46,38);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,24)(20,33)(21,32)(22,37)(23,31)(25,40)(26,29)(27,44)(28,39)(30,35)(34,47)(36,46)(38,42)(41,45)(43,48);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,38)(26,43)(27,41)(28,42)(29,45)(31,33)(32,47)(37,46)(39,40)(44,48);(9,11,12,16)(10,14,13,15)(19,31)(24,39)(29,44)(30,33)(32,37)(38,40)(45,48)(46,47);(17,21,23,22)(18,26,28,27)(19,32,31,37)(20,34,35,36)(24,29,39,44)(25,41,42,43)(30,46,33,47)(38,48,40,45)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=hi,e^2=i,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, 6371073);