#6623843

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#2867	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1580 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)(3,7)(4,6)(5,8)(10,14)(15,16)(17,18,23,26)(19,27,28,32)(20,21,25,22)(24,29,31,30);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19,21,30,23,28,22,29)(18,24,20,27,26,31,25,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20,22,18,23,25,21,26)(19,27,29,24,28,32,30,31);(1,4,5,3)(2,6,8,7)(11,12)(15,16)(19,28)(24,31)(27,32)(29,30);(9,12,13,11)(10,16,14,15)(17,21,23,22)(18,25,26,20)(19,29,28,30)(24,27,31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,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, 6531039);