#347445

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(3,9)(4,11)(6,14)(7,12)(10,13)(15,16)(18,24)(19,29)(20,32)(21,22)(26,41)(27,28)(31,35)(33,34)(36,39)(38,46)(43,44)(47,48);(1,2,5,8)(3,6,10,15)(4,12,11,7)(9,16,13,14)(17,18)(19,25)(20,26)(21,27)(22,24)(23,28)(29,39)(30,38)(31,40)(32,37)(33,42)(34,41)(35,36)(43,47)(44,46)(45,48);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,11)(12,14)(17,19)(18,25)(20,30)(21,29)(22,35)(23,31)(24,36)(26,38)(27,39)(28,40)(32,44)(33,43)(34,45)(37,46)(41,48)(42,47);(17,20,23,34)(18,26,28,41)(19,30,31,45)(21,33,22,32)(24,37,27,42)(25,38,40,48)(29,43,35,44)(36,46,39,47)

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,g^a=gh,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254641);