#417103

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

762 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 242 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324299);