#5035205

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 2097152
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

1730 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=f,c^2=fgi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef

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, 4942401);