#7514195

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

3096 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)(5,6,11,17)(7,18,20,25)(8,13,24,32)(9,14,12,26)(10,16,27,15)(19,33,21,34)(22,29,28,35)(23,31,36,30);(1,2)(3,4)(5,7,11,20)(6,12,17,9)(8,21,24,19)(10,26,27,14)(13,22,32,28)(15,25,16,18)(23,29,36,35)(30,33,31,34);(2,4)(5,8,11,24)(6,13,17,32)(7,19,20,21)(9,22,12,28)(10,23,27,36)(14,29,26,35)(15,30,16,31)(18,33,25,34);(2,4)(5,9,11,12)(6,14,17,26)(7,15,20,16)(8,22,24,28)(10,25,27,18)(13,29,32,35)(19,30,21,31)(23,34,36,33);(5,9)(6,7)(8,22)(10,18)(11,12)(13,19)(14,16)(15,26)(17,20)(21,32)(23,33)(24,28)(25,27)(29,31)(30,35)(34,36)

Polycyclic presentation:

2^9|a^2=i,b^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, 7421391);