#122998

order	512
nilpotency class	4
isoclinism	isoclinic class of #122991
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

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

353 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,6)(4,13)(5,9)(7,10)(8,11)(12,15)(14,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,44)(30,43)(31,42)(32,38)(33,39)(34,41)(35,45)(36,46)(37,47)(40,48);(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)(17,19,22,30)(18,23,26,36)(20,31,33,28)(21,29,34,46)(24,37,39,35)(25,27,41,43)(32,47,48,45)(38,42,40,44);(17,20,22,33)(18,24,26,39)(19,28,30,31)(21,32,34,48)(23,35,36,37)(25,40,41,38)(27,42,43,44)(29,45,46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30194);