#415628

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gh,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322824);