#940594

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1932 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:

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

Polycyclic presentation:

2^9|a^2=fghi,c^2=hi,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 847790);