#6467544

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2329 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)(13,14,17,16,18,21,26,19)(15,20,22,25,24,27,28,23);(2,4)(14,19)(15,22)(16,21)(17,26)(23,25)(24,28);(6,9)(10,12)(13,15,17,22,18,24,26,28)(14,20,16,25,21,27,19,23);(13,16,17,21,18,19,26,14)(15,23,22,20,24,25,28,27);(5,7,8,11)(6,10,9,12)(15,24)(20,27)(22,28)(23,25)

Polycyclic presentation:

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

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, 6374740);