#4600851

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1781 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)(5,6,8,10)(7,11,12,9)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30)(31,37)(32,36)(33,34)(35,38)(39,44)(40,43)(41,42);(5,6,8,10)(7,9,12,11)(15,22)(18,21)(23,28)(25,27)(30,34)(31,38)(36,42)(39,40);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,31)(30,35)(32,40)(33,38)(34,37)(36,44)(39,41)(42,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,22)(14,18,20,21)(16,23,24,28)(19,25,26,27)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,44)(41,42);(5,7,8,12)(6,9,10,11)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,32,33,41)(30,36,34,42)(31,39,38,40)(35,43,37,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4508047);