#7614099

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=h,e^2=f,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,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, 7521295);