#6453803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3357 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:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,18)(17,21)(19,22)(20,24)(25,27)(26,28);(1,2)(3,4)(13,15)(14,20)(16,23)(17,24)(18,26)(19,25)(21,27)(22,28);(2,4)(6,9)(10,12)(13,16,19,21)(14,17,22,18)(15,23,25,27)(20,24,28,26);(2,4)(13,14)(15,20)(16,17)(18,21)(19,22)(23,24)(25,28)(26,27);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,22,16)(15,24,25,26)(20,27,28,23)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6360999);