#6552726

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#38033	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1886 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,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^c=di,e^d=ehi

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, 6459922);