#6490052

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1359 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,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, 6397248);