#6458410

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,26)(21,27)(22,37)(23,28)(24,32)(25,35)(30,44)(31,40)(33,45)(34,42)(36,41)(38,48)(39,46)(43,47);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,31,23,33,22,32)(18,24,37,45,28,40,27,29)(20,30,36,48,35,47,34,46)(25,38,41,44,26,39,42,43);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,38)(26,28)(27,42)(29,43)(31,48)(32,46)(33,47)(37,41)(39,40)(44,45);(11,15)(13,16)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,39)(25,28)(27,41)(29,44)(31,48)(32,46)(33,47)(37,42)(38,40)(43,45);(17,21,23,22)(18,27,28,37)(19,31,33,32)(20,34,35,36)(24,29,40,45)(25,41,26,42)(30,46,47,48)(38,44,39,43)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6365606);