#6512884

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3216 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)(13,14,20,17)(15,23,27,28)(16,26,24,25)(18,21,19,22);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,20,27)(14,16,17,24)(18,26,19,25)(21,23,22,28);(6,9)(7,11)(13,16)(14,15)(17,27)(18,28)(19,23)(20,24)(21,25)(22,26);(5,7,8,11)(6,10,9,12)(13,17,20,14)(15,24,27,16)(18,21,19,22)(23,25,28,26);(6,9)(10,12)(13,18,20,19)(14,21,17,22)(15,25,27,26)(16,28,24,23)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6420080);