#263498

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	#62342	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

839 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,d^2=ehi,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, 170694);