#7514158

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

3082 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)(5,6,11,16)(7,15,10,12)(8,19,22,13)(9,23,24,30)(14,25,29,18)(17,26,21,28)(20,33,34,36)(27,31,32,35);(1,2)(3,4)(6,12)(7,10)(8,17)(14,23)(15,16)(18,25)(20,31)(21,22)(26,28)(29,30)(33,36)(34,35);(2,4)(5,7,11,10)(6,12,16,15)(8,17,22,21)(9,18,24,25)(13,26,19,28)(14,23,29,30)(20,31,34,35)(27,36,32,33);(2,4)(5,8,11,22)(6,13,16,19)(7,17,10,21)(9,20,24,34)(12,26,15,28)(14,27,29,32)(18,31,25,35)(23,36,30,33);(5,9)(6,14)(7,18)(8,20)(10,25)(11,24)(12,23)(13,27)(15,30)(16,29)(17,31)(19,32)(21,35)(22,34)(26,36)(28,33)

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,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, 7421354);