#7514170

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514160
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

3079 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,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,f^b=fi,h^b=hi,e^d=ei,h^d=hi,f^e=fi

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, 7421366);