#3497911

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2428 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:

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

Polycyclic presentation:

2^9|b^2=fghi,d^2=fi,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, 3405107);