#332869

order	512
nilpotency class	3
isoclinism	isoclinic class of #332839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18916	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

700 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240065);