#4344480

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1761 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)(5,6,10,12)(7,8,9,11)(15,21)(16,23)(19,24)(20,25)(29,30,34,35)(31,32,38,36)(33,41,42,37)(39,43,44,40);(5,7,10,9)(6,8,12,11)(14,18)(19,24)(20,25)(26,28)(30,35)(31,38)(33,42)(40,43);(5,8,10,11)(6,7,12,9)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,38)(30,36,35,32)(33,39,42,44)(37,43,41,40);(1,2,3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,36)(30,31,35,38)(33,40,42,43)(37,39,41,44);(5,9,10,7)(6,11,12,8)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,42)(30,37,35,41)(31,39,38,44)(32,40,36,43)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,d^2=ghi,e^2=fgh,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, 4251676);