#263832

order	512
nilpotency class	3
isoclinism	isoclinic class of #263819
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	#62414	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

848 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)(3,10)(4,14)(5,8)(6,9)(7,11)(12,15)(13,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,27)(28,31)(29,32);(1,2,5,14,6,9,4,8)(3,7,12,16,13,15,11,10)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(1,2)(3,7)(4,8)(5,14)(6,9)(10,11)(12,16)(13,15)(17,19,22,29)(18,23,25,31)(20,28,21,27)(24,32,30,26);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171028);