#6453938

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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

3160 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)(7,10)(8,9)(11,12)(13,14,19,24)(15,26,21,16)(17,22,28,27)(18,25,20,23);(2,4)(13,15)(14,17)(16,23)(18,22)(19,21)(20,27)(24,28)(25,26);(6,9)(10,12)(13,16)(14,21)(15,24)(17,20)(18,28)(19,26)(22,25)(23,27);(13,17,19,28)(14,22,24,27)(15,23,21,25)(16,20,26,18);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,23,24,25)(15,22,21,27)(16,28,26,17)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361134);