#6464008

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437748
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371204);