#351356

order	512
nilpotency class	3
isoclinism	isoclinic class of #347698
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

652 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)(3,5)(4,6)(7,8)(10,13)(11,14)(15,16)(17,18,23,28)(19,26,30,31)(20,25,22,27)(21,24,29,32);(1,3,4,7)(2,5,6,8)(17,19,20,21,23,30,22,29)(18,24,27,26,28,32,25,31);(2,6)(5,8)(9,10,12,13)(11,14,15,16)(17,20,23,22)(18,25,28,27)(19,21,30,29)(24,31,32,26);(9,11)(10,14)(12,15)(13,16)(17,21,20,30,23,29,22,19)(18,26,27,32,28,31,25,24)

Polycyclic presentation:

2^9|a^2=h,b^2=efhi,c^2=i,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,d^b=dh,g^d=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258552);