#6464154

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6371350);