#6471294

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,33,41)(20,26,35,42)(21,27,22,37)(25,36,43,34)(29,44,30,39)(31,40,32,47)(38,46,48,45);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,31)(22,32)(23,33)(25,38)(26,44)(27,40)(28,41)(30,35)(34,45)(36,46)(37,47)(39,42)(43,48);(11,15)(13,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,38)(26,37)(27,42)(28,43)(30,33)(31,45)(32,46)(39,40)(41,48)(44,47);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,39)(25,37)(27,43)(28,42)(29,33)(31,45)(32,46)(38,40)(41,44)(47,48);(10,14)(13,16)(17,21,23,22)(18,27,28,37)(19,31,33,32)(20,34,35,36)(24,40,41,47)(25,42,43,26)(29,45,30,46)(38,39,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,g^c=gi,e^d=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, 6378490);