#6464371

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1589 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)(17,18,23,21)(19,27,30,32)(20,26,22,25)(24,28,31,29);(9,10)(11,15)(12,13)(14,16)(17,19,23,30)(18,24,21,31)(20,29,22,28)(25,27,26,32);(2,5)(6,8)(10,13)(14,15)(17,20,23,22)(18,25,21,26)(19,28,30,29)(24,32,31,27);(9,11,12,16)(10,14,13,15)(17,21,23,18)(19,24,30,31)(20,26,22,25)(27,28,32,29);(1,3,4,7)(2,6,5,8)(17,20,23,22)(18,26,21,25)(19,29,30,28)(24,32,31,27)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371567);