#129442

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

520 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,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, 36638);