#6433713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
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	#29606	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 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)(17,18,22,28)(19,29,20,24)(21,27,23,26)(25,30,32,31);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,22)(21,31)(23,30)(25,26)(27,32)(28,29);(11,15)(13,16)(17,20,22,19)(18,25,28,32)(21,31,23,30)(24,27,29,26);(10,14)(13,16)(17,21,22,23)(18,26,28,27)(19,30,20,31)(24,32,29,25);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,28,26)(19,30,20,31)(24,25,29,32)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^b=ei,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340909);