#6617639

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3069 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)(13,14,19,21,18,22,17,20)(15,23,24,16,26,27,25,28);(6,10)(8,11)(13,15)(14,16)(17,24)(18,26)(19,25)(20,27)(21,23)(22,28);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,23)(18,28)(19,27)(20,25)(21,24)(22,26);(5,7,9,12)(6,11,10,8)(13,17,18,19)(14,20,22,21)(15,24,26,25)(16,27,28,23);(5,8,9,11)(6,7,10,12)(13,17,18,19)(14,21,22,20)(15,24,26,25)(16,23,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524835);