#6578073

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 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)(8,12)(9,10)(13,14,20,17)(15,18,22,19)(16,23,26,24)(21,28,27,25);(1,2)(3,4)(6,10)(11,12)(13,15,20,22)(14,18,17,19)(16,24,26,23)(21,28,27,25);(2,4)(13,16,20,26)(14,21,17,27)(15,23,22,24)(18,25,19,28);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,17,20,14)(15,18,22,19)(16,21,26,27)(23,28,24,25);(2,4)(5,8,9,7)(6,12,10,11)(13,18,20,19)(14,22,17,15)(16,25,26,28)(21,24,27,23)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,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

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485269);