#352384

order	512
nilpotency class	3
isoclinism	isoclinic class of #352384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

672 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,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,c^b=ci,d^b=di,d^c=di,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, 259580);