#317192

order	512
nilpotency class	3
isoclinism	isoclinic class of #317171
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

521 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,d^c=di,f^c=fi,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, 224388);