#7003533

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

2036 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)(15,17)(20,22)(21,24)(23,26)(27,28);(5,6)(7,11)(8,10)(9,12)(13,14,17,19,18,22,15,20)(16,24,23,21,25,27,26,28);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,17)(14,20,22,19)(16,23,25,26)(21,27,28,24);(7,12)(9,11)(13,16,18,25)(14,21,22,28)(15,23,17,26)(19,24,20,27);(6,10)(9,11)(14,22)(16,25)(19,20)(23,26)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^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,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, 6910729);