#322421

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	#19676	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

727 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)(4,6)(5,9)(7,12)(8,11)(13,14)(15,16)(17,18,22,26,23,27,21,30)(19,25,28,31,29,32,20,24);(2,6,9,8)(7,14,15,10)(17,19,21,20,23,29,22,28)(18,24,26,25,27,31,30,32);(1,2)(3,7)(4,8)(5,9)(6,11)(10,12)(13,15)(14,16)(17,18)(19,24)(20,25)(21,26)(22,30)(23,27)(28,32)(29,31);(1,3,11,16,5,13,4,12)(2,7,6,14,9,15,8,10)(17,20,22,19,23,28,21,29)(18,25,30,24,27,32,26,31)

Polycyclic presentation:

2^9|a^2=f,b^2=eghi,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, 229617);