#6584888

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#25762	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2017 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6492084);