#321980

order	512
nilpotency class	3
isoclinism	isoclinic class of #321957
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,f^c=fi,g^c=gh

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, 229176);