#5481495

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

1822 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)(7,11)(8,10)(9,12)(14,18)(15,21)(16,24)(26,28)(29,30,34,31)(32,37,40,35)(33,36,41,38)(39,43,44,42);(5,7)(6,9)(8,12)(10,11)(14,18)(16,24)(19,22)(23,27)(30,31)(33,41)(35,37)(39,44);(6,10)(7,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,31,34,30)(32,37,40,35)(33,36,41,38)(39,42,44,43);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32,34,40)(30,35,31,37)(33,39,41,44)(36,42,38,43);(5,6)(7,9)(8,10)(11,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,33)(30,36)(31,38)(32,39)(34,41)(35,42)(37,43)(40,44)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=fg,e^2=fh,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, 5388691);