#3363320

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=gh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

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, 3270516);