#116924

order	512
nilpotency class	3
isoclinism	isoclinic class of #116918
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

533 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=ehi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24120);