#3299593

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

1483 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fgh,d^2=fhi,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, 3206789);