#489317

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2965 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

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, 396513);