#6438429

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂×#3045	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,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, 6345625);