#417077

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

758 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 218 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=ef,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=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, 324273);