#6627416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6534612);