#3325124

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1460 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)(17,18,21,22)(19,25,26,23)(20,24,28,30)(27,32,29,31)(33,34)(35,40)(36,43)(37,39)(38,45)(41,46)(42,48)(44,47);(10,14)(15,16)(18,22)(19,26)(20,28)(31,32)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,2,5,4)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,3,5,7)(2,6,4,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,29)(21,26)(22,25)(24,32)(27,28)(30,31)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,16,14,15)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=fi,d^2=hi,e^2=fghi,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, 3232320);