#7563334

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

2689 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=g,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, 7470530);