#239350

order	512
nilpotency class	3
isoclinism	isoclinic class of #239347
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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

836 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18)(19,27)(20,31)(21,35)(22,25)(23,29)(24,33)(26,37)(28,44)(30,42)(32,43)(34,41)(36,39)(38,47)(40,45)(46,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19,22,30)(18,23,25,39)(20,32,33,28)(21,36,37,29)(24,40,31,38)(26,42,35,27)(34,45,48,47)(41,43,46,44);(4,11)(6,12)(7,14)(9,15)(17,20,21,34,22,33,37,48)(18,24,26,41,25,31,35,46)(19,28,29,45,30,32,36,47)(23,38,27,43,39,40,42,44);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(18,25)(20,33)(23,39)(26,35)(27,42)(28,32)(34,48)(45,47)

Polycyclic presentation:

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

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, 146546);