#7054570

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2164 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)(15,22)(16,26)(17,18)(19,28)(20,23)(21,25)(24,27);(5,6)(7,11)(8,10)(9,12)(13,15,20,16)(14,21,23,24)(17,26,28,22)(18,25,19,27);(2,4)(5,7,8,12)(6,9,10,11)(13,16,20,15)(14,22,23,26)(17,24,28,21)(18,25,19,27);(7,12)(9,11)(13,17,20,28)(14,18,23,19)(15,24,16,21)(22,27,26,25);(6,10)(9,11)(13,18,20,19)(14,17,23,28)(15,25,16,27)(21,26,24,22)

Polycyclic presentation:

2^9|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=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961766);