#332598

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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,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, 239794);