#4134778

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=fg,d^2=fg,e^2=fg,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, 4041974);