#124539

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cgi,d^a=df,c^b=ci,d^b=dg,e^b=efg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31735);