#129440

order	512
nilpotency class	4
isoclinism	isoclinic class of #129431
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	C₂×#117	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 510 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)(19,29,33,45)(20,26,21,35)(22,27)(23,28)(24,31)(25,34,42,36)(30,41)(32,39,37,48)(38,43)(40,47)(44,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24,28,41)(20,30)(21,31)(22,37)(23,33)(25,39,42,48)(26,29,35,45)(27,44,43,40)(32,38)(34,47)(36,46);(3,9)(6,12)(10,15)(13,16)(17,20)(18,25,28,42)(19,30)(21,23)(22,34)(24,39,41,48)(26,43,35,27)(29,40,45,44)(31,33)(32,46)(36,38)(37,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,e^b=ei,f^b=fgi,d^c=di,f^c=fi,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, 36636);