#7502153

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500736
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=df,f^b=fi,g^c=gi,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, 7409349);