#7514194

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

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,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, 7421390);