#5475192

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

1843 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)(7,11)(10,12)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(29,30,34,37)(31,38,32,35)(33,41,42,36)(39,43,40,44);(6,9)(10,12)(14,15)(17,25)(19,21)(23,28)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,21,24,19)(17,20,25,22)(23,26,28,27)(29,30)(31,35)(32,38)(33,41)(34,37)(36,42)(39,44)(40,43);(1,2)(3,4)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(5,7)(6,10)(8,11)(9,12)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,36,37,41)(31,39,32,40)(35,43,38,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=f,e^2=g,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, 5382388);