#4145528

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

1735 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)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,27,26,28)(29,30,34,32)(31,37,39,35)(33,40,41,36)(38,42,44,43);(5,7,10,9)(6,11,8,12)(14,18)(16,25)(19,21)(22,26)(33,41)(36,40)(38,44)(42,43);(5,8,10,6)(7,11,9,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,9,10,7)(6,11,8,12)(13,15,17,23)(14,19,18,21)(16,26,25,22)(20,28,24,27)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(13,16,17,25)(14,20,18,24)(15,22,23,26)(19,27,21,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=gh,d^2=fgh,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, 4052724);