#349085

order	512
nilpotency class	3
isoclinism	isoclinic class of #349070
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3480	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=i,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,d^b=di,f^c=fi,g^d=gh

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, 256281);