#6475508

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1738 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,34)(21,25)(22,37)(23,27)(24,32)(26,29)(30,44)(31,41)(33,42)(35,40)(36,48)(38,46)(39,43)(45,47);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,29,43,23,33,22,32)(18,24,37,42,27,39,26,28)(20,30,36,46,21,31,35,45)(25,38,48,44,34,47,40,41);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(17,20,23,21)(18,25,27,34)(19,30,33,31)(22,35,29,36)(24,38,39,47)(26,40,37,48)(28,41,42,44)(32,45,43,46);(11,15)(13,16)(17,21,23,20)(18,25,27,34)(19,31,33,30)(22,36,29,35)(24,38,39,47)(26,40,37,48)(28,41,42,44)(32,46,43,45);(10,14)(13,16)(20,21)(25,34)(30,31)(35,36)(38,47)(40,48)(41,44)(45,46)

Polycyclic presentation:

2^9|b^2=fg,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,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, 6382704);