#489094

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61284	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

709 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

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, 396290);