#6579138

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
prime radical	C₂×#120	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2123 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)(7,11)(8,10)(9,12)(13,14)(15,18)(16,20)(17,21)(19,24)(22,26)(23,27)(25,28);(5,7,9,8)(6,10,12,11)(13,15,17,24)(14,18,21,19)(16,22,25,23)(20,26,28,27);(2,4)(5,7,9,8)(6,11,12,10)(15,22)(16,25)(18,26)(19,27)(20,28)(23,24);(5,7,9,8)(6,11,12,10)(13,15)(14,19)(16,22)(17,24)(18,21)(20,27)(23,25)(26,28);(5,8,9,7)(6,10,12,11)(15,22)(16,25)(18,26)(19,27)(20,28)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308602
maximal normal subgroup of	#72308602

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486334);