#6460183

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,31,25)(20,22,27,24)(21,26,30,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,25,28,29)(20,26,27,32)(21,24,30,22);(10,14)(11,15)(17,20,24,18,23,27,22,28)(19,26,30,25,31,32,21,29);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,21,23,30)(18,26,28,32)(19,22,31,24)(20,29,27,25);(17,22,23,24)(18,27,28,20)(19,30,31,21)(25,26,29,32)

Polycyclic presentation:

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

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, 6367379);