#6439140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2168 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)(5,6,9,11)(7,12,8,10)(13,14)(15,21)(16,22)(17,25)(18,20)(19,23)(24,27)(26,28);(5,7,9,8)(6,10,11,12)(14,20)(15,24)(16,21)(17,19)(22,26)(27,28);(2,4)(5,8,9,7)(6,10,11,12)(13,15,18,26)(14,21,20,28)(16,25,27,23)(17,24,19,22);(13,16)(14,22)(15,25)(17,21)(18,27)(19,28)(20,24)(23,26);(13,17,18,19)(14,23,20,25)(15,22,26,24)(16,21,27,28)

Polycyclic presentation:

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

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, 6346336);