#6458566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 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)(4,8)(5,6)(9,10,13,15)(11,14,12,16)(17,18,23,28,24,30,22,29)(19,31,34,25,36,42,35,43)(20,26,37,44,38,45,21,27)(32,47,33,40,39,41,48,46);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,33)(21,39)(22,34)(23,35)(24,36)(26,41)(27,46)(28,31)(29,42)(30,43)(32,37)(38,48)(40,44)(45,47);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,22,21,24,38,23,37)(18,26,29,27,30,45,28,44)(19,32,35,48,36,39,34,33)(25,40,31,47,43,46,42,41);(1,4,5,3)(2,8,6,7)(17,21,22,38,24,37,23,20)(18,27,29,45,30,44,28,26)(19,33,35,32,36,48,34,39)(25,41,31,40,43,47,42,46);(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,21,38,37)(25,31,43,42)(26,44,45,27)(32,33,39,48)(40,47,46,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365762);