#6472911

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	#37418	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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)(11,13)(12,14)(15,16)(17,18)(19,29)(20,35)(21,26)(22,39)(23,28)(24,33)(25,32)(27,30)(31,44)(34,42)(36,40)(37,41)(38,48)(43,46)(45,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,30,37,23,34,22,33)(18,24,39,42,28,41,27,29)(20,31,46,38,36,21,32,47)(25,26,40,48,43,44,35,45);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,32,23,36,30,46)(18,25,27,35,28,43,39,40)(19,31,33,47,34,21,37,38)(24,26,29,45,41,44,42,48);(11,15)(13,16)(17,21,23,31)(18,26,28,44)(19,32,34,46)(20,33,36,37)(22,38,30,47)(24,40,41,35)(25,42,43,29)(27,45,39,48);(10,14)(13,16)(20,36)(21,31)(25,43)(26,44)(32,46)(35,40)(38,47)(45,48)

Polycyclic presentation:

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

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, 6380107);