#133741

order	512
nilpotency class	4
isoclinism	isoclinic class of #133293
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

633 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,32,46,22,57)(3,15,19,49,53,64,56,39)(4,10,5,25,28,45,58,44)(7,14,33,40,8,13,27,47)(9,34,37,60,62,21,55,17)(11,23,42,29,59,24,41,30)(16,50,26,38,54,63,52,36)(18,48,51,35,20,43,31,61);(1,3,8,20)(2,9,13,38)(4,21,24,16)(5,26,30,17)(6,31,27,19)(7,18,32,53)(10,39,41,35)(11,43,25,15)(12,37,47,63)(14,36,46,62)(22,51,33,56)(23,54,28,34)(29,60,58,52)(40,50,57,55)(42,61,45,49)(44,64,59,48);(1,4,8,24)(2,10,13,41)(3,16,20,21)(5,27,30,6)(7,23,32,28)(9,35,38,39)(11,12,25,47)(14,42,46,45)(15,37,43,63)(17,19,26,31)(18,34,53,54)(22,58,33,29)(36,49,62,61)(40,59,57,44)(48,50,64,55)(51,60,56,52)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgi,d^c=dh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40937);