#6570593

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2163 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,d^2=hi,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,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, 6477789);