#6464068

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1922 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)(14,19)(15,23)(16,17)(21,22)(25,26)(27,28);(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,22,18,21,16,19)(15,24,23,27,25,20,26,28);(6,9)(10,11)(13,15,17,23,18,25,16,26)(14,20,22,28,21,24,19,27);(5,7,8,12)(6,10,9,11)(14,21)(19,22)(20,24)(27,28);(15,25)(20,24)(23,26)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371264);