#6597171

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#37657	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2285 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)(7,10)(8,11)(9,12)(13,14,19,16)(15,22,26,28)(17,21,18,27)(20,24,23,25);(5,7)(6,9)(8,12)(10,11)(13,15,18,25,19,26,17,24)(14,20,21,22,16,23,27,28);(2,4)(7,12)(9,10)(13,14)(15,23)(16,19)(17,27)(18,21)(20,26)(22,24)(25,28);(5,7)(6,10)(8,12)(9,11)(13,16,19,14)(15,20,26,23)(17,21,18,27)(22,24,28,25);(7,12)(9,10)(13,17,19,18)(14,21,16,27)(15,24,26,25)(20,22,23,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh

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, 6504367);