#491144

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=h,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,c^b=ci,d^c=di,f^c=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, 398340);