#6584247

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

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, 6491443);