#332642

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=hi,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=de,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239838);