#133801

order	512
nilpotency class	4
isoclinism	isoclinic class of #133422
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, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2312 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=eh,f^b=fg,d^c=dh,e^c=ei,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, 40997);