#6459798

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3365 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:

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

Polycyclic presentation:

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

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, 6366994);