#4455160

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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

1809 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4362356);