#6577826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#36022	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2069 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)(5,6)(7,11)(8,12)(9,10)(13,14,18,19)(15,23,26,20)(16,21,27,25)(17,22,24,28);(6,10)(11,12)(13,15,18,26)(14,20,19,23)(16,17,27,24)(21,28,25,22);(2,4)(14,19)(15,23)(16,25)(20,26)(21,27)(22,28);(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,24)(17,26)(18,27)(19,25)(20,22)(23,28);(5,8,9,7)(6,12,10,11)(13,17,18,24)(14,22,19,28)(15,25,26,21)(16,20,27,23)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485022);