#7614104

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563341
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

2536 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)(7,18)(8,13)(9,25)(10,16)(11,17)(12,20)(14,24)(15,27)(19,31)(21,30)(22,33)(23,36)(26,32)(28,35)(29,34);(2,3)(5,7,10,20,11,22,27,21)(6,12,15,18,17,30,16,33)(8,23,9,19,24,35,26,34)(13,31,32,36,14,29,25,28);(1,2,4,3)(8,24)(9,26)(13,14)(19,34)(23,35)(25,32)(28,36)(29,31);(1,3,4,2)(5,8,10,9,11,24,27,26)(6,13,16,25,17,14,15,32)(7,19,21,23,22,34,20,35)(12,28,18,31,30,36,33,29);(5,9,10,24,11,26,27,8)(6,14,15,25,17,13,16,32)(7,19,20,35,22,34,21,23)(12,29,18,28,30,31,33,36)

Polycyclic presentation:

2^9|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,d^c=di,e^c=ei,e^d=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, 7521300);