#7514182

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
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

3050 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:

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

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=cgi,d^b=dh,f^b=fi,h^b=hi,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, 7421378);