#5035981

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 3145728
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

1431 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fg,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4943177);