#241112

order	512
nilpotency class	3
isoclinism	isoclinic class of #241105
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

842 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148308);