#234441

order	512
nilpotency class	3
isoclinism	isoclinic class of #234438
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	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 141637);