#6595236

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,20)(21,24)(22,28)(23,25)(26,27)(29,31)(30,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,20,25,28)(21,26,30,31)(24,27,32,29);(11,15)(13,16)(17,20,19,18,23,28,22,25)(21,29,26,32,30,27,31,24);(1,3,5,4)(2,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,21)(18,24)(19,26)(20,27)(22,31)(23,30)(25,32)(28,29);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(21,30)(24,32)(26,31)(27,29)

Polycyclic presentation:

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

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, 6502432);