#3217747

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=fgh,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,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, 3124943);