#7502154

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

2652 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:

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

Polycyclic presentation:

2^9|c^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, 7409350);