#4161440

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

1716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=gh,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4068636);