#7054746

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×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2176 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,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=ehi,f^b=fi,d^c=dh,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, 6961942);