#4142848

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=h,d^2=gh,e^2=g,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, 4050044);