#132783

order	512
nilpotency class	4
isoclinism	isoclinic class of #132559
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	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

503 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fghi,d^c=dghi,f^c=fg,e^d=ei,f^d=fi,f^e=fi

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, 39979);