#1226605

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
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

1590 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fi,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1133801);