#129489

order	512
nilpotency class	4
isoclinism	isoclinic class of #129437
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 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2274 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

539 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=di,f^b=fgi,e^c=ei,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36685);