#6461726

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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)(4,8)(5,7)(17,18,22,26)(19,27,28,42)(20,24,32,40)(21,34,36,25)(23,35,37,30)(29,43,31,45)(33,48,47,41)(38,46,39,44);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,35)(22,28)(24,39)(25,42)(26,37)(27,34)(29,32)(30,36)(33,44)(38,40)(41,43)(45,48)(46,47);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(19,28)(23,37)(27,42)(29,31)(30,35)(38,39)(43,45)(44,46);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,20)(18,24)(19,29)(21,33)(22,32)(23,38)(25,41)(26,40)(27,43)(28,31)(30,44)(34,48)(35,46)(36,47)(37,39)(42,45);(20,32)(24,40)(29,31)(33,47)(38,39)(41,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6368922);