#311347

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 218543);