#7514171

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514160
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

3065 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)(5,6,12,18)(7,19,16,30)(8,23,28,14)(9,22,10,15)(11,21,13,17)(20,34,24,33)(25,31,26,36)(27,32,29,35);(1,2)(3,4)(5,7,12,16)(6,10,18,9)(8,24,28,20)(11,15,13,22)(14,25,23,26)(17,30,21,19)(27,36,29,31)(32,33,35,34);(2,4)(5,8,12,28)(6,14,18,23)(7,20,16,24)(9,25,10,26)(11,27,13,29)(15,31,22,36)(17,32,21,35)(19,33,30,34);(2,4)(5,9,12,10)(6,15,18,22)(7,21,16,17)(8,25,28,26)(11,30,13,19)(14,31,23,36)(20,35,24,32)(27,34,29,33);(5,10)(6,16)(7,18)(8,26)(9,12)(11,30)(13,19)(14,24)(15,21)(17,22)(20,23)(25,28)(27,34)(29,33)(31,35)(32,36)

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=cg,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, 7421367);