#3192671

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

1744 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgh,c^2=ghi,d^2=ghi,e^2=f,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, 3099867);