#6576840

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	#35750	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2805 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)(7,12)(8,10)(9,11)(13,14,18,16)(15,23,26,20)(17,21,25,24)(19,22,28,27);(5,7,9,8)(6,10,11,12)(13,15,18,26)(14,20,16,23)(17,19,25,28)(21,27,24,22);(2,4)(13,16,18,14)(15,24,26,21)(17,20,25,23)(19,22,28,27);(5,8,9,7)(6,10,11,12)(13,17)(14,21)(15,19)(16,24)(18,25)(20,27)(22,23)(26,28);(15,25)(17,26)(19,28)(20,21)(22,27)(23,24)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=ehi,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, 6484036);