#263467

order	512
nilpotency class	3
isoclinism	isoclinic class of #263459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19332	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 170663);