#6452112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3509 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=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, 6359308);