#7614106

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

2494 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)(7,17)(8,14)(9,24)(10,25)(11,15)(12,16)(13,19)(18,32)(20,29)(21,27)(22,31)(23,35)(26,30)(28,33)(34,36);(2,3)(5,7,10,19,12,20,11,26)(6,13,15,17,16,30,25,29)(8,21,23,33,22,34,9,18)(14,28,24,27,31,32,35,36);(1,2,4,3)(8,22)(9,23)(14,31)(18,33)(21,34)(24,35)(27,36)(28,32);(1,3,4,2)(5,8,11,9,12,22,10,23)(6,14,15,24,16,31,25,35)(7,18,19,21,20,33,26,34)(13,27,29,28,30,36,17,32);(5,9,11,22,12,23,10,8)(6,14,25,35,16,31,15,24)(7,18,26,34,20,33,19,21)(13,28,29,36,30,32,17,27)

Polycyclic presentation:

2^9|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,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, 7521302);