#6617844

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	#39130	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1858 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)(13,14,20,24)(15,18,23,19)(16,21,27,25)(17,22,28,26);(6,10)(8,11)(13,15,19,14,20,23,18,24)(16,26,17,25,27,22,28,21);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,21)(15,22)(17,19)(18,28)(20,27)(23,26)(24,25);(5,7,9,12)(6,11,10,8)(13,17)(14,22)(15,25)(16,18)(19,27)(20,28)(21,23)(24,26);(5,8,9,11)(6,7,10,12)(13,18,20,19)(14,23,24,15)(16,17,27,28)(21,22,25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6525040);