#6994334

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2348 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)(15,23)(16,19)(17,18)(20,22)(21,24)(25,26)(27,28);(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,23)(17,26)(18,24)(19,27)(20,25)(22,28);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,19)(15,23)(17,20)(18,22)(21,27)(24,28)(25,26);(7,12)(9,11)(13,17,20,16)(14,18,22,19)(15,21,25,28)(23,24,26,27);(6,10)(9,11)(13,18,20,19)(14,17,22,16)(15,24,25,27)(21,26,28,23)

Polycyclic presentation:

2^9|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,f^b=fi,d^c=dhi,e^c=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, 6901530);