#4602177

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

1923 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)(15,18)(16,21)(17,20)(19,23)(22,25)(24,27)(26,28)(33,41)(36,40)(38,44)(42,43);(5,6)(7,9)(8,10)(11,12)(14,15)(16,23)(20,22)(24,28)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(7,12)(9,11)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(5,7)(6,9)(8,12)(10,11)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=h,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, 4509373);