#6466971

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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

1450 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)(19,24)(20,25)(21,23)(22,26)(27,38)(28,35)(29,37)(30,33)(31,40)(32,34)(36,42)(39,43)(41,47)(44,45)(46,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,23)(19,27)(20,31)(22,28)(24,36)(25,39)(29,44)(30,42)(32,43)(33,38)(34,40)(46,47);(9,12,13,11)(10,14,15,16)(17,19,28,42,21,30,22,27)(18,24,35,36,23,33,26,38)(20,29,43,48,32,44,31,41)(25,37,39,46,34,45,40,47);(17,20,22,31,21,32,28,43)(18,25,26,40,23,34,35,39)(19,29,27,41,30,44,42,48)(24,37,38,47,33,45,36,46);(1,3,4,7)(2,6,5,8)(20,32)(25,34)(29,44)(31,43)(37,45)(39,40)(41,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374167);