#5083064

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2389 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22,26,27)(23,28,29,24)(25,30,34,35)(31,36,33,32);(8,12)(10,11)(13,15,18,16)(14,19,20,17)(22,27)(24,28)(30,35)(32,36);(6,7)(10,11)(14,20)(15,16)(21,23)(22,28)(24,27)(25,33)(26,29)(30,32)(31,34)(35,36);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,19,20,17)(21,24,26,28)(22,29,27,23)(25,32,34,36)(30,33,35,31);(5,8)(6,10)(7,11)(9,12)(13,17,18,19)(14,15,20,16)(21,25)(22,30)(23,31)(24,32)(26,34)(27,35)(28,36)(29,33)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4990260);