#334803

order	512
nilpotency class	3
isoclinism	isoclinic class of #334695
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	#3245	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241999);