#6465352

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1339 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=h,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,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, 6372548);