#4668358

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1760 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)(6,10)(7,11)(13,14,18,17)(15,16,20,19)(21,22,27,32)(23,29,24,28)(25,34,35,30)(26,33,36,31);(5,6)(7,8)(9,10)(11,12)(13,15)(14,19)(16,17)(18,20)(21,23,27,24)(22,28,32,29)(25,33,35,31)(26,30,36,34);(6,10)(8,12)(14,17)(15,20)(21,24,27,23)(22,29,32,28)(25,33,35,31)(26,30,36,34);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,25,27,35)(22,30,32,34)(23,31,24,33)(26,29,36,28);(5,8)(6,7)(9,12)(10,11)(13,17,18,14)(15,16,20,19)(21,26,27,36)(22,31,32,33)(23,30,24,34)(25,29,35,28)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fh,d^2=hi,e^2=g,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, 4575554);