#6569302

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535648
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fi,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^d=ehi

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, 6476498);