#3854773

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,4)(3,7,8,6)(17,18,21,22)(19,23,26,30)(20,24,29,28)(25,31,27,32)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(10,14)(15,16)(18,22)(23,30)(24,28)(31,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,2,5,4)(3,6,8,7)(9,10,13,14)(11,15,12,16)(17,19,21,26)(18,23,22,30)(20,27,29,25)(24,32,28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,15,14,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,32)(26,30)(27,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,16,14,15)(17,20,21,29)(18,24,22,28)(19,25,26,27)(23,31,30,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=fghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3761969);