#6660413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660408
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	#35251	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2479 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6567609);