#4975879

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 2097152
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1843 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=ghi,c^2=i,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4883075);