#352270

order	512
nilpotency class	3
isoclinism	isoclinic class of #352255
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

655 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)(3,10)(4,7)(9,14)(11,12)(13,15)(17,18,21,24)(19,25,27,32)(20,29,26,23)(22,30,31,28);(1,2,7,14,5,9,4,6)(3,8,10,15,12,16,11,13);(17,19,26,30,21,27,20,28)(18,22,29,32,24,31,23,25);(1,3)(2,8)(4,11)(5,12)(6,13)(7,10)(9,16)(14,15)(19,27)(22,31)(25,32)(28,30)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,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,d^c=di,g^d=gh

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, 259466);