#5475506

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

1752 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)(5,6,9,7)(8,10,12,11)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(30,34)(31,37)(36,40)(39,43);(6,7)(8,12)(15,22)(16,25)(18,21)(19,24)(29,30,33,34)(31,35,37,38)(32,36,41,40)(39,42,43,44);(5,7,9,6)(8,10,12,11)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,30)(31,38)(32,40)(33,34)(35,37)(36,41)(39,42)(43,44);(1,2)(3,4)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,44)(41,43);(5,8,9,12)(6,10,7,11)(13,16)(14,19)(15,23)(17,25)(18,26)(20,24)(21,27)(22,28)(29,32,33,41)(30,36,34,40)(31,39,37,43)(35,42,38,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=gh,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, 5382702);