#3364419

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2254 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)(6,9)(7,11)(13,14)(15,19)(16,18)(17,20)(21,22)(23,29)(24,28)(25,26)(27,30)(31,36)(32,34)(33,35);(6,9)(10,12)(13,15,18,17)(14,19,16,20)(22,26)(23,30)(28,32)(31,33);(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,20,16,19)(21,22)(23,27)(24,32)(25,26)(28,34)(29,30)(31,36)(33,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,19,17,20)(21,23,25,30)(22,27,26,29)(24,33,34,31)(28,36,32,35);(5,7)(6,10)(8,11)(9,12)(13,17,18,15)(14,19,16,20)(21,24)(22,28)(23,31)(25,34)(26,32)(27,35)(29,36)(30,33)

Polycyclic presentation:

2^9|b^2=fgh,c^2=fgh,d^2=gh,e^2=fgh,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, 3271615);