#4861897

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1981 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)(6,11)(7,8)(15,19)(18,20)(21,22,26,27)(23,28,30,24)(25,33,35,29)(31,32,36,34);(5,6,10,11)(7,9,8,12)(14,17)(18,20)(22,27)(23,30)(29,33)(31,36);(5,7,10,8)(6,9,11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,27)(25,31)(26,30)(29,34)(32,33)(35,36);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(21,24,26,28)(22,23,27,30)(25,34,35,32)(29,36,33,31);(5,9)(6,8)(7,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,35)(22,29,27,33)(23,31,30,36)(24,32,28,34)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=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, 4769093);