#6443056

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(5,6,10,13,11,15,22,14)(7,16,19,28,20,27,18,9)(8,21,25,12,26,29,24,30)(17,32,33,36,35,23,34,31);(1,2)(3,4)(5,7)(6,9)(8,17)(10,18)(11,20)(12,23)(13,27)(14,16)(15,28)(19,22)(21,31)(24,33)(25,34)(26,35)(29,36)(30,32);(2,4)(5,8,22,24,11,26,10,25)(6,12,13,29,15,30,14,21)(7,17,19,33,20,35,18,34)(9,23,27,36,28,32,16,31);(2,4)(5,9)(6,7)(8,23)(10,16)(11,28)(12,17)(13,19)(14,18)(15,20)(21,34)(22,27)(24,36)(25,31)(26,32)(29,33)(30,35);(5,10,11,22)(6,13,15,14)(7,18,20,19)(8,24,26,25)(9,27,28,16)(12,21,30,29)(17,33,35,34)(23,31,32,36)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,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, 6350252);