#6483031

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

1456 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)(20,23)(22,26)(24,27)(25,30)(28,31)(29,32);(2,5)(6,8)(10,14)(11,12)(17,19,23,30,24,31,22,29)(18,25,26,21,27,32,20,28);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,23,18,24,26,22,27)(19,28,30,25,31,21,29,32);(9,12,13,11)(10,16,14,15)(17,21)(18,19)(20,29)(22,25)(23,32)(24,28)(26,30)(27,31);(10,14)(15,16)(17,22,24,23)(18,26,27,20)(19,29,31,30)(21,32,28,25)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=h,e^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=dhi,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, 6390227);