#4934450

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=f,e^2=g,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, 4841646);