#132810

order	512
nilpotency class	4
isoclinism	isoclinic class of #132541
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)(17,18,22,28,23,26,21,27)(19,24,31,38,30,39,35,42)(20,32,33,25,34,40,36,41)(29,43,46,47,44,48,45,37);(1,2)(3,7)(4,11)(5,8)(6,13)(9,14)(10,16)(12,15)(17,19)(18,24,26,39)(20,29)(21,35)(22,31)(23,30)(25,37,41,47)(27,38,28,42)(32,43,40,48)(33,46)(34,44)(36,45);(1,3,5,10)(2,7,8,16)(4,12,13,9)(6,14,11,15)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,41)(27,32)(28,40)(30,44)(31,45)(35,46)(38,43)(39,47)(42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=egh,f^b=fghi,d^c=dghi,f^c=fgi,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, 40006);