#348082

order	512
nilpotency class	3
isoclinism	isoclinic class of #348082
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	#18995	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

866 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255278);