#129622

order	512
nilpotency class	4
isoclinism	isoclinic class of #129617
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator, prime radical	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

537 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=chi,e^b=egi,f^b=fg,d^c=dg,f^c=fi,e^d=ei,f^d=fi

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, 36818);