#7514156

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514148
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

3041 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)(7,17)(8,20)(9,24)(10,16)(11,12)(13,22)(14,25)(15,28)(18,23)(19,21)(26,31)(27,29)(30,32)(33,36)(34,35);(1,2)(3,4)(5,7,11,18)(6,8,12,22)(9,19,26,33)(10,13,28,20)(14,21,29,36)(15,23,16,17)(24,32,31,35)(25,34,27,30);(2,4)(6,12)(7,18)(13,20)(14,29)(15,16)(19,33)(24,31)(30,34);(2,4)(5,8,11,22)(6,13,12,20)(7,16,18,15)(9,21,26,36)(10,17,28,23)(14,30,29,34)(19,24,33,31)(25,35,27,32);(5,9)(6,14)(7,19)(8,21)(10,27)(11,26)(12,29)(13,30)(15,31)(16,24)(17,32)(18,33)(20,34)(22,36)(23,35)(25,28)

Polycyclic presentation:

2^9|b^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,h^d=hi,f^e=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421352);