#321910

order	512
nilpotency class	3
isoclinism	isoclinic class of #321901
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

698 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229106);