#7614096

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563327
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

2592 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:

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

Polycyclic presentation:

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

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, 7521292);