#488818

order	512
nilpotency class	4
isoclinism	isoclinic class of #488707
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
prime radical	#33783	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

693 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 693 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396014);