#132849

order	512
nilpotency class	4
isoclinism	isoclinic class of #132509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#56558	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 533 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)(19,28)(20,25,34,44)(21,26,36,24)(22,27)(23,30,39,32)(29,42)(31,45,35,48)(33,41)(37,43)(38,47)(40,46);(1,3)(2,7)(4,15)(5,12)(6,13)(8,9)(10,16)(11,14)(17,19)(18,23)(20,32)(21,35)(22,29)(24,40)(25,42)(26,38)(27,39)(28,44)(30,34)(31,36)(33,46)(37,47)(41,48)(43,45);(1,3)(2,8)(4,12)(5,15)(6,13)(7,9)(10,11)(14,16)(18,24)(19,29)(21,36)(23,38)(25,43)(26,27)(28,45)(30,32)(33,37)(39,40)(41,44)(42,48)

Polycyclic presentation:

2^9|a^2=f,d^2=h,e^2=i,f^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=eghi,f^b=fgh,d^c=dgh,e^c=ei,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, 40045);