#4314400

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

1736 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)(6,10)(7,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(30,31)(32,39)(36,38)(40,44);(15,22)(18,21)(23,28)(26,27)(29,30,34,31)(32,35,39,37)(33,38,41,36)(40,43,44,42);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,31,34,30)(32,35,39,37)(33,38,41,36)(40,42,44,43);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(5,8,9,11)(6,12,10,7)(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=fgh,e^2=g,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, 4221596);