#234517

order	512
nilpotency class	3
isoclinism	isoclinic class of #232291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,e^b=eh,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141713);