#6438452

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2238 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)(5,6,8,9)(7,10,11,12)(13,14)(15,22)(16,19)(17,27)(18,21)(20,25)(23,26)(24,28);(6,9)(10,12)(13,15,18,25)(14,20,21,22)(16,23,27,28)(17,26,19,24);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,27)(14,17,21,19)(15,22,25,20)(23,26,28,24);(13,14)(15,23)(16,17)(18,21)(19,27)(20,24)(22,26)(25,28);(13,17,18,19)(14,16,21,27)(15,24,25,26)(20,28,22,23)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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,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, 6345648);