#6483043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,21)(20,22)(23,27)(24,26)(25,28)(29,32)(30,31);(2,5)(6,8)(10,14)(11,12)(17,19,23,29,24,30,22,28)(18,25,20,31,26,32,27,21);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,18,24,27,23,26)(19,21,28,32,30,31,29,25);(9,12,13,11)(10,16,14,15)(17,21)(18,19)(20,29)(22,32)(23,25)(24,31)(26,30)(27,28);(10,14)(15,16)(18,26)(20,27)(21,31)(25,32)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=hi,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=dhi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390239);