#332768

order	512
nilpotency class	3
isoclinism	isoclinic class of #332711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24227	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

670 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)(3,6)(4,7)(5,8)(9,10,15,20)(11,21,25,32)(12,26,29,18)(13,16,14,19)(17,22,33,24)(23,35,37,40)(27,34,28,30)(31,36,39,38);(1,3,5,4)(2,6,8,7)(10,16)(11,22)(12,27)(13,14)(19,20)(21,32)(23,37)(24,25)(28,29)(30,34)(31,40)(35,39);(2,7)(3,4)(6,8)(9,11)(10,17)(12,23)(13,22)(14,24)(15,25)(16,21)(18,31)(19,32)(20,33)(26,39)(27,36)(28,38)(29,37)(30,35)(34,40);(2,7)(3,4)(6,8)(9,12,15,29)(10,18,20,26)(11,23,25,37)(13,27,14,28)(16,30,19,34)(17,31,33,39)(21,35,32,40)(22,36,24,38)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=dei,e^b=eh,f^b=fh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239964);