#248404

order	512
nilpotency class	3
isoclinism	isoclinic class of #248359
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25741	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1144 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fhi,d^c=di,f^c=fi,e^d=eh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155600);