#7563336

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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

2753 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:

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

Polycyclic presentation:

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

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, 7470532);