#7003386

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2113 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6910582);