#3330516

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

1384 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=fhi,d^2=ghi,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, 3237712);