#322412

order	512
nilpotency class	3
isoclinism	isoclinic class of #322409
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19671	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

709 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229608);