#488624

order	512
nilpotency class	4
isoclinism	isoclinic class of #488539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

730 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:

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

Polycyclic presentation:

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

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, 395820);