#6579189

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486385);