#347417

order	512
nilpotency class	3
isoclinism	isoclinic class of #347322
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62926	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2024 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,e^d=eh,g^d=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254613);