#5475509

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1890 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)(6,10)(7,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,34,37)(31,35,32,39)(33,36,41,44)(38,42,40,43);(5,6)(7,11)(8,12)(9,10)(15,22)(16,25)(18,21)(19,24)(29,31,34,32)(30,35,37,39)(33,40,41,38)(36,43,44,42);(6,10)(8,11)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,32,34,31)(30,35,37,39)(33,40,41,38)(36,42,44,43);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,30)(31,35)(32,39)(33,36)(34,37)(38,42)(40,43)(41,44);(5,8,9,11)(6,7,10,12)(13,16)(14,19)(15,23)(17,25)(18,26)(20,24)(21,27)(22,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,44)(39,43)

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=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, 5382705);