#6435391

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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)(11,15)(14,16)(17,18)(19,25)(20,33)(21,27)(22,28)(23,38)(24,29)(26,34)(30,40)(31,41)(32,43)(35,45)(36,44)(37,42)(39,47)(46,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,30)(21,35)(22,37)(23,31)(24,32)(26,39)(27,45)(28,42)(29,43)(33,40)(34,47)(36,46)(38,41)(44,48);(10,13)(14,16)(17,20,24,21)(18,26,29,44)(19,30,32,35)(22,34,23,36)(25,39,43,48)(27,38,33,28)(31,46,37,47)(40,42,45,41);(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,27,29,33)(19,30,32,35)(22,34,23,36)(25,40,43,45)(26,38,44,28)(31,47,37,46)(39,42,48,41);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,28,29,38)(19,31,32,37)(20,34,21,36)(25,41,43,42)(26,27,44,33)(30,46,35,47)(39,40,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342587);