#6598537

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1897 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)(13,14,20,15)(16,26,27,28)(17,22,21,25)(18,23,19,24);(5,6,8,9)(7,10,11,12)(13,15,18,23,20,14,19,24)(16,21,25,28,27,17,22,26);(2,4)(6,9)(7,11)(13,16,18,25,20,27,19,22)(14,21,24,28,15,17,23,26);(5,7)(6,10)(8,11)(9,12)(13,17)(14,22)(15,25)(16,24)(18,26)(19,28)(20,21)(23,27);(6,9)(7,11)(13,18,20,19)(14,23,15,24)(16,25,27,22)(17,28,21,26)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=f,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^b=di,e^b=eh,d^c=dh,e^d=eh

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, 6505733);