#4145506

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

1816 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)(5,6,10,8)(7,11,9,12)(13,14,18,16)(15,21,23,19)(17,24,25,20)(22,26,28,27)(29,30)(31,37)(32,41)(33,43)(34,35)(36,40)(38,39)(42,44);(5,7,10,9)(6,11,8,12)(17,25)(20,24)(22,28)(26,27)(30,35)(33,41)(36,37)(39,44);(5,8,10,6)(7,11,9,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,31)(30,36)(32,42)(33,44)(34,40)(35,37)(38,43)(39,41);(1,2)(3,4)(5,9,10,7)(6,11,8,12)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,32,34,43)(30,33,35,41)(31,38,40,42)(36,39,37,44);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,33)(30,32)(31,39)(34,41)(35,43)(36,38)(37,42)(40,44)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=fgh,d^2=gh,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, 4052702);