#352309

order	512
nilpotency class	3
isoclinism	isoclinic class of #352300
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

658 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)(7,13)(10,11)(14,16)(17,18,21,24)(19,25,28,32)(20,29,26,23)(22,30,31,27);(1,2,4,6,5,7,12,13)(3,8,10,14,11,15,9,16);(2,7)(6,13)(8,15)(14,16)(17,19,26,30,21,28,20,27)(18,22,29,32,24,31,23,25);(1,3)(2,8)(4,10)(5,11)(6,14)(7,15)(9,12)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259505);