#6994300

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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

2359 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)(14,19)(16,17)(20,22)(21,23)(24,26)(27,28);(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,19)(17,22)(18,20)(21,26)(24,27)(25,28);(2,4)(5,7)(6,9)(8,12)(10,11)(14,20)(19,22)(21,27)(23,28);(7,12)(9,11)(13,15)(14,21)(16,24)(17,26)(18,25)(19,23)(20,27)(22,28);(6,10)(9,11)(13,16,18,17)(14,22,20,19)(15,24,25,26)(21,28,27,23)

Polycyclic presentation:

2^9|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,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, 6901496);