#232692

order	512
nilpotency class	3
isoclinism	isoclinic class of #232688
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
prime radical	#60948	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1228 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,45)(32,40)(33,39)(34,38)(35,42)(36,44)(37,43)(41,46)(47,48);(6,15)(9,16)(17,19,23,33)(18,24,28,40)(20,34,36,47)(21,32,22,31)(25,41,44,45)(26,29,27,39)(30,37,46,35)(38,43,48,42);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,42)(27,43)(28,44)(29,45)(31,34)(32,47)(33,46)(39,41)(40,48);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,42,44,43)(30,34,46,47)(38,41,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139888);