#352810

order	512
nilpotency class	3
isoclinism	isoclinic class of #352174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=di,d^c=dh

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, 260006);