#314609

order	512
nilpotency class	3
isoclinism	isoclinic class of #251791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2482
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

511 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18,22,27,23,28,21,26)(19,24,30,39,31,40,37,44)(20,32,36,25,34,43,35,42)(29,41,33,48,45,47,46,38);(2,7)(8,15)(9,10)(11,16)(17,19)(18,24)(20,33,34,46)(21,37)(22,30)(23,31)(25,41,42,47)(26,44)(27,39)(28,40)(29,36,45,35)(32,38,43,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,25)(19,29)(21,36)(22,35)(23,34)(24,38)(26,43)(27,32)(28,42)(30,46)(31,45)(33,37)(39,47)(40,48)(41,44);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(17,19)(18,24)(20,29)(21,37)(22,30)(23,31)(25,38)(26,44)(27,39)(28,40)(32,47)(33,36)(34,45)(35,46)(41,43)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fhi,g^c=gh,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 221805);