#326669

order	512
nilpotency class	3
isoclinism	isoclinic class of #251809
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

477 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 233865);