#7514174

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

3129 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)(5,6,10,13)(7,17,21,33)(8,23,25,14)(9,26,11,16)(12,22,28,20)(15,27,31,24)(18,32,35,34)(19,30,36,29);(1,2)(3,4)(5,7,10,21)(6,12,13,28)(8,19,25,36)(9,22,11,20)(14,32,23,34)(15,30,31,29)(16,33,26,17)(18,27,35,24);(2,4)(6,13)(12,28)(14,23)(15,31)(16,26)(17,33)(29,30)(32,34);(2,4)(5,8)(6,14)(7,18)(9,24)(10,25)(11,27)(12,29)(13,23)(15,16)(17,34)(19,20)(21,35)(22,36)(26,31)(28,30)(32,33);(5,8)(6,15)(7,19)(9,27)(10,25)(11,24)(12,30)(13,31)(14,16)(17,34)(18,20)(21,36)(22,35)(23,26)(28,29)(32,33)

Polycyclic presentation:

2^9|a^2=i,b^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, 7421370);