#132858

order	512
nilpotency class	4
isoclinism	isoclinic class of #132521
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×#117	C₂³
prime radical	#2815	C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

535 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,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, 40054);