#6456365

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363561);