#132489

order	512
nilpotency class	4
isoclinism	isoclinic class of #132309
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2356 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 461 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,f^c=fg,f^d=fi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39685);