#6432788

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
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×#121	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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,26)(21,35)(22,27)(23,28)(24,29)(30,39)(31,43)(32,41)(33,42)(34,44)(36,40)(37,45)(38,46)(47,48);(2,5)(6,8)(17,19)(18,25)(20,31)(21,36)(22,32)(23,38)(24,33)(26,43)(27,41)(28,46)(29,42)(30,34)(35,40)(37,47)(39,44)(45,48);(10,13)(14,16)(17,20,24,21)(18,26,29,35)(19,30,33,47)(22,34,23,37)(25,39,42,48)(27,44,28,45)(31,32,36,38)(40,46,43,41);(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,26,29,35)(19,31,33,36)(22,34,23,37)(25,40,42,43)(27,45,28,44)(30,32,47,38)(39,46,48,41);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,27,29,28)(19,32,33,38)(20,34,21,37)(25,41,42,46)(26,44,35,45)(30,36,47,31)(39,40,48,43)

Polycyclic presentation:

2^9|c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339984);