#7514151

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514148
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

3085 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)(7,17)(8,21)(9,19)(10,16)(11,15)(12,25)(13,26)(14,28)(18,34)(20,27)(22,33)(23,30)(24,31)(29,35)(32,36);(1,2)(3,4)(5,7)(6,9)(8,22)(10,13)(11,20)(12,29)(14,27)(15,26)(16,17)(18,25)(19,28)(21,23)(24,35)(30,36)(31,33)(32,34);(2,4)(5,8,11,25)(6,12,15,21)(7,18,20,22)(9,23,26,29)(10,24,28,36)(13,30,19,35)(14,31,16,32)(17,33,27,34);(2,4)(5,9)(6,13)(7,14)(8,23)(10,17)(11,26)(12,30)(15,19)(16,20)(18,31)(21,35)(22,32)(24,33)(25,29)(27,28)(34,36);(5,9)(6,7)(8,23)(10,27)(11,26)(12,18)(13,14)(15,20)(16,19)(17,28)(21,22)(24,34)(25,29)(30,31)(32,35)(33,36)

Polycyclic presentation:

2^9|c^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,h^d=hi,f^e=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421347);