#6597159

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	#37656	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2243 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)(15,24)(16,26)(17,21)(18,22)(19,25)(20,23)(27,28);(5,7)(6,9)(8,12)(10,11)(13,15,18,26,20,27,19,21)(14,17,25,28,23,16,22,24);(2,4)(7,12)(9,10)(13,16)(14,21)(15,25)(17,20)(18,24)(19,28)(22,27)(23,26);(5,7)(6,10)(8,12)(9,11)(13,17,20,16)(14,21,23,26)(15,22,27,25)(18,28,19,24);(7,12)(9,10)(13,18,20,19)(14,22,23,25)(15,26,27,21)(16,28,17,24)

Polycyclic presentation:

2^9|b^2=fh,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, 6504355);