#6584767

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,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=eh,d^c=dhi,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, 6491963);