#1151645

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fgi,d^2=fgh,e^2=fg,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, 1058841);