#133111

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	#42652	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#540 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 720 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)(19,28)(20,25,34,42)(21,24)(22,27)(23,29,38,32)(26,35)(30,43)(31,40)(33,39,48,41)(36,45,37,47)(44,46);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19,22,31)(18,23,27,38)(20,32,34,29)(21,30,35,46)(24,36,26,37)(25,40,42,28)(33,47,48,45)(39,44,41,43);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(18,24)(21,35)(23,36)(25,39)(26,27)(28,43)(30,46)(33,48)(37,38)(40,44)(41,42)(45,47)

Polycyclic presentation:

2^9|a^2=f,b^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, 40307);