#6470995

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6378191);