#235420

order	512
nilpotency class	3
isoclinism	isoclinic class of #235411
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142616);