#347887

order	512
nilpotency class	3
isoclinism	isoclinic class of #347878
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

1026 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,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, 255083);