#245883

order	512
nilpotency class	3
isoclinism	isoclinic class of #245807
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	#24227	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1104 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)(11,21)(12,25)(13,19)(14,17)(15,20)(16,23)(18,27)(22,31)(24,33)(26,35)(28,36)(29,34)(30,32)(37,38)(39,40);(1,2,5,7)(3,6,4,8)(9,11,15,24)(10,16,20,32)(12,26,29,22)(13,30,14,23)(17,33,19,21)(18,35,36,31)(25,40,34,37)(27,38,28,39);(1,3,5,4)(2,6,7,8)(10,17)(13,14)(16,21)(18,34)(19,20)(23,30)(25,36)(27,28)(31,37)(32,33)(35,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,12,15,29)(10,18,20,36)(11,22,24,26)(13,28,14,27)(16,31,32,35)(17,34,19,25)(21,37,33,40)(23,39,30,38)

Polycyclic presentation:

2^9|b^2=h,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,d^b=dh,e^b=eh,e^c=ei,f^c=fi,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, 153079);