#1581112

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2182 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gh,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 1488308);