#6468833

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	#34609	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1907 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)(17,18,23,25,22,19,21,26)(20,27,30,32,28,24,29,31);(9,11)(10,13)(12,15)(14,16)(17,19,22,18)(20,27,28,24)(21,26,23,25)(29,32,30,31);(2,5)(6,8)(10,14)(11,15)(17,20,22,28)(18,24,19,27)(21,29,23,30)(25,31,26,32);(11,15)(13,16)(20,28)(24,27)(29,30)(31,32);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,25,19,26)(20,29,28,30)(24,31,27,32)

Polycyclic presentation:

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

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, 6376029);