#249800

order	512
nilpotency class	3
isoclinism	isoclinic class of #249799
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	#60650	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1348 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)(9,10)(11,20)(12,25)(13,22)(14,33)(15,18)(16,19)(17,29)(21,38)(23,32)(24,31)(26,37)(27,39)(28,36)(30,35)(34,40);(3,7)(5,8)(9,11)(10,13)(12,26)(14,23)(15,22)(16,24)(17,34)(18,31)(19,32)(20,33)(21,28)(25,27)(29,39)(30,40)(35,38)(36,37);(1,2)(3,8)(4,6)(5,7)(9,12,16,30)(10,17,19,36)(11,21,24,39)(13,27,32,38)(14,28,15,29)(18,35,33,25)(20,37,31,34)(22,26,23,40);(3,7)(5,8)(9,13)(10,11)(12,27)(14,31)(15,20)(16,32)(17,21)(18,23)(19,24)(22,33)(25,26)(28,34)(29,37)(30,38)(35,40)(36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156996);