#4138134

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

1783 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)(13,14,18,19)(15,16,21,24)(17,20,27,28)(22,23,25,26)(29,30)(31,35)(32,39)(33,41)(34,37)(36,42)(38,44)(40,43);(6,9)(10,12)(14,19)(15,21)(20,28)(22,25)(29,31,34,32)(30,35,37,39)(33,40,42,38)(36,44,41,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,25)(18,21)(19,24)(20,26)(22,27)(23,28)(29,32,34,31)(30,35,37,39)(33,40,42,38)(36,43,41,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,26)(18,24)(19,21)(20,25)(22,28)(23,27)(29,30)(31,35)(32,39)(33,36)(34,37)(38,43)(40,44)(41,42);(13,17,18,27)(14,20,19,28)(15,22,21,25)(16,23,24,26)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,41)(39,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4045330);