#2027673

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1681 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)(3,6)(4,5)(7,8)(10,12)(11,13)(17,18,21,24)(19,22,20,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(2,4)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,20)(18,22,24,23)(26,27)(29,37)(31,33)(35,40);(1,2,5,4)(3,8,7,6)(9,11,14,13)(10,15,12,16)(17,19,21,20)(18,22,24,23)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,3,5,7)(2,6,4,8)(9,12,14,10)(11,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(17,20,21,19)(18,23,24,22)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ghi,c^2=fghi,d^2=f,e^2=hi,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, 1934869);