#132806

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

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

normal subgroups only

This group has 502 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)(19,28,31,41)(20,32,35,39)(21,26,36,25)(22,27)(23,30)(24,34)(29,37)(33,40)(38,46)(42,43,47,44)(45,48);(1,2)(3,7)(4,11)(5,8)(6,13)(9,14)(10,16)(12,15)(17,19)(18,23,27,38)(20,29)(21,30)(22,31)(24,37,40,48)(25,28,26,41)(32,47,39,42)(33,43)(34,44)(35,45)(36,46);(1,3,5,10)(2,7,8,16)(4,12,13,9)(6,14,11,15)(17,20,22,35)(18,24,27,40)(19,29,31,45)(21,34,36,33)(23,37,38,48)(25,39,26,32)(28,42,41,47)(30,44,46,43)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=hi,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, 40002);