#6458144

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1404 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)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,25)(21,23)(22,35)(24,31)(26,29)(28,42)(30,41)(32,37)(33,46)(34,48)(36,43)(38,45)(39,47)(40,44);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(20,32)(26,35)(31,43)(34,44)(38,39)(41,46);(11,15)(13,16)(17,19,29,43,21,28,22,31)(18,24,35,42,23,36,26,27)(20,33,44,45,32,30,34,47)(25,39,48,41,37,38,40,46);(11,15)(13,16)(17,20)(18,25)(19,30)(21,32)(22,34)(23,37)(24,38)(26,40)(27,41)(28,33)(29,44)(31,45)(35,48)(36,39)(42,46)(43,47);(1,3,4,7)(2,6,5,8)(19,28)(20,32)(24,36)(25,37)(27,42)(31,43)(34,44)(40,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=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, 6365340);