#6605899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#38089	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1629 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)(11,13)(12,14)(15,16)(17,18,23,21)(19,28,31,25)(20,24,32,30)(22,26,29,27);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,29,32,23,31,22,20)(18,24,26,25,21,30,27,28);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,29,19,23,32,22,31)(18,25,26,30,21,28,27,24);(9,11)(10,13)(12,15)(14,16)(17,21,23,18)(19,30,31,24)(20,25,32,28)(22,26,29,27);(11,15)(13,16)(17,22,23,29)(18,26,21,27)(19,20,31,32)(24,25,30,28)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^d=eh

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, 6513095);