#132634

order	512
nilpotency class	4
isoclinism	isoclinic class of #132549
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×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 512 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) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,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, 39830);