#347935

order	512
nilpotency class	3
isoclinism	isoclinic class of #347908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255131);