#5249623

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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

1500 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5156819);