#6468924

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=gh,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6376120);