#6465138

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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

1367 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)(4,6)(5,8)(17,18,23,28)(19,29,31,41)(20,34,38,43)(21,27,22,26)(24,33,40,32)(25,36,44,37)(30,39,35,42)(45,46,48,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,21,32,23,31,22,33)(18,24,26,29,28,40,27,41)(20,35,36,47,38,30,37,46)(25,42,43,48,44,39,34,45);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,21,36,23,38,22,37)(18,25,26,43,28,44,27,34)(19,30,32,46,31,35,33,47)(24,39,29,45,40,42,41,48);(9,11,12,16)(10,14,13,15)(19,31)(24,40)(29,41)(30,35)(32,33)(39,42)(45,48)(46,47);(1,4,5,3)(2,6,8,7)(19,31)(24,40)(29,41)(30,35)(32,33)(39,42)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=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, 6372334);