#133110

order	512
nilpotency class	4
isoclinism	isoclinic class of #133109
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18)(19,27,30,39)(20,31,33,23)(21,25)(22,26)(24,29,38,34)(28,42)(32,46,48,37)(35,40)(36,43)(41,44)(45,47);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(18,23)(24,36)(25,37)(26,31)(27,41)(38,45)(39,42)(40,46);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(17,19,22,30)(18,24,26,38)(20,28,33,44)(21,34,35,29)(23,36,31,45)(25,39,40,27)(32,47,48,43)(37,42,46,41)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,f^b=fgh,e^c=ei,f^c=fhi,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40306);