#4937296

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1854 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)(5,6,9,11)(7,10,8,12)(15,19)(18,20)(21,22)(23,24)(25,29)(26,27)(28,30)(31,32)(33,34)(35,36);(5,7,9,8)(6,10,11,12)(14,17)(18,20)(22,27)(23,30)(25,33)(32,35);(6,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,27)(25,31)(26,30)(29,35)(32,34)(33,36);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(21,24,26,28)(22,23,27,30)(25,32,33,35)(29,31,34,36);(5,8,9,7)(6,12,11,10)(13,15,16,19)(14,18,17,20)(21,25,26,33)(22,29,27,34)(23,31,30,36)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4844492);