#3497166

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

2431 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=fg,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, 3404362);