#6576584

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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

2920 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)(7,12)(8,10)(9,11)(13,14,20,16)(15,21,17,22)(18,23,19,24)(25,28,27,26);(5,7,9,8)(6,10,11,12)(13,15,20,17)(14,21,16,22)(18,26,19,28)(23,25,24,27);(2,4)(13,16,20,14)(15,25,17,27)(18,23,19,24)(21,28,22,26);(5,8,9,7)(6,10,11,12)(13,17,20,15)(14,22,16,21)(18,28,19,26)(23,27,24,25);(13,18)(14,23)(16,24)(19,20)(25,27)(26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6483780);