#6467029

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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)(9,10,13,15)(11,14,12,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,47)(31,42)(33,46)(34,40)(35,43)(36,44)(37,45)(38,41)(39,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,34)(21,30)(22,32)(23,38)(25,42)(26,39)(27,29)(28,46)(31,35)(33,37)(36,48)(40,43)(41,45)(44,47);(9,12,13,11)(10,14,15,16)(17,20,23,21,22,35,37,36)(18,25,28,26,27,43,45,44)(19,30,33,34,32,48,38,31)(24,39,41,42,29,47,46,40);(17,21,23,35,22,36,37,20)(18,26,28,43,27,44,45,25)(19,31,33,30,32,34,38,48)(24,40,41,39,29,42,46,47);(1,3,4,7)(2,6,5,8)(18,27)(24,29)(25,43)(26,44)(28,45)(39,47)(40,42)(41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6374225);