#123018

order	512
nilpotency class	4
isoclinism	isoclinic class of #123011
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	#3186	C₂²
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

323 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 42 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=eh,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30214);