#6579413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36122	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2076 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

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, 6486609);