#232872

order	512
nilpotency class	3
isoclinism	isoclinic class of #232862
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1205 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)(4,6)(5,8)(9,13)(10,16)(11,14)(12,15)(17,18)(19,29)(20,34)(21,27)(22,26)(23,28)(24,32)(25,35)(30,38)(31,40)(33,39)(36,43)(37,42)(41,46)(44,45)(47,48);(2,6)(3,10)(4,13)(8,9)(11,12)(14,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,41)(26,39)(27,29)(28,40)(34,44)(35,45)(36,47)(37,46)(38,43)(42,48);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,16)(10,13)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,34)(27,42)(28,43)(29,44)(31,45)(32,47)(33,46)(39,48)(40,41);(2,8)(6,9)(7,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,39,40,29)(25,34,43,42)(30,45,46,47)(38,48,41,44)

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140068);