#2575817

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1563 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)(10,13)(11,12)(17,18,21,20)(19,22,23,24)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,40)(38,39);(9,10,14,13)(11,16,12,15)(19,23)(22,24)(26,31)(28,35)(33,39)(36,40);(9,11,14,12)(10,15,13,16)(17,19,21,23)(18,22,20,24)(25,27,30,32)(26,28,31,35)(29,37,38,34)(33,40,39,36);(1,2,4,5)(3,6,7,8)(9,12,14,11)(10,15,13,16)(17,18,21,20)(19,22,23,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,37)(34,39)(38,40);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,15,12,16)(17,20,21,18)(19,24,23,22)(25,29,30,38)(26,33,31,39)(27,34,32,37)(28,36,35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fgh,d^2=fhi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2483013);