#245786

order	512
nilpotency class	3
isoclinism	isoclinic class of #245711
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1188 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)(3,4)(7,8)(9,10)(11,21)(12,25)(13,19)(14,17)(15,20)(16,23)(18,26)(22,30)(24,32)(27,34)(28,33)(29,31)(35,36)(37,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,22)(13,29)(14,23)(15,24)(17,32)(18,30)(19,21)(20,31)(25,35)(26,38)(27,36)(28,37)(33,39)(34,40);(1,3,5,4)(2,6,7,8)(10,17)(13,14)(16,21)(18,33)(19,20)(23,29)(25,34)(26,27)(30,35)(31,32)(36,38)(39,40);(1,2)(3,6)(4,8)(5,7)(9,12,15,28)(10,18,20,34)(11,22,24,37)(13,27,14,26)(16,30,31,40)(17,33,19,25)(21,35,32,39)(23,38,29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152982);