#110136

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17332);