#4342873

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

1682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6,10,9)(7,12,11,8)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,26,28,27)(33,41)(36,40)(38,44)(42,43);(14,15)(17,25)(19,21)(23,28)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,7)(6,8)(9,12)(10,11)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(5,9,10,6)(7,12,11,8)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=h,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4250069);