#6451728

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4132 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)(7,12)(8,10)(9,11)(13,14,20,17)(15,23,26,28)(16,18,22,19)(21,27,24,25);(1,2)(3,4)(13,15)(14,21)(16,23)(17,24)(18,25)(19,27)(20,26)(22,28);(2,4)(5,7,9,8)(6,10,11,12)(13,16)(14,19)(15,23)(17,18)(20,22)(21,27)(24,25)(26,28);(2,4)(5,8,9,7)(6,10,11,12)(13,17,20,14)(15,24,26,21)(16,18,22,19)(23,25,28,27);(13,18,20,19)(14,22,17,16)(15,25,26,27)(21,28,24,23)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,e^2=i,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, 6358924);