#4669541

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2171 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)(5,6,11,10)(7,8,12,9)(15,19)(18,20)(21,22)(23,28)(24,33)(25,26)(27,30)(29,36)(31,35)(32,34);(5,7)(6,9)(8,10)(11,12)(14,17)(18,20)(21,23,27,24)(22,28,30,33)(25,34,36,31)(26,32,29,35);(5,8,11,9)(6,12,10,7)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,28,30,33)(25,34,36,31)(26,35,29,32);(1,2,3,4)(5,9,11,8)(6,12,10,7)(21,25)(22,29)(23,31)(24,34)(26,30)(27,36)(28,32)(33,35);(5,10,11,6)(7,8,12,9)(13,15,16,19)(14,18,17,20)(21,26,27,29)(22,25,30,36)(23,32,24,35)(28,34,33,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fg,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,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, 4576737);