#134186

order	512
nilpotency class	4
isoclinism	isoclinic class of #133912
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2316 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

502 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,f^b=fg,e^c=ei,f^c=fhi,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, 41382);