#7518806

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2518 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426002);