#321991

order	512
nilpotency class	3
isoclinism	isoclinic class of #321983
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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

686 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)(7,13)(8,12)(11,14)(15,16)(17,18,20,25,21,24,28,23)(19,22,26,31,27,30,32,29);(2,6,9,8)(3,11)(7,10,15,14)(13,16)(17,19,21,27)(18,22,24,30)(20,26,28,32)(23,29,25,31);(1,2)(3,7)(4,8)(5,9)(6,12)(10,13)(11,15)(14,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,3,12,16,5,11,4,13)(2,7,6,14,9,15,8,10)(18,24)(22,30)(23,25)(29,31)

Polycyclic presentation:

2^9|a^2=f,b^2=eghi,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,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, 229187);