#235513

order	512
nilpotency class	3
isoclinism	isoclinic class of #235504
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1124 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)(3,13)(4,18)(5,11)(6,10)(7,12)(8,15)(9,19)(14,25)(16,24)(17,23)(20,27)(21,26)(22,30)(28,29)(31,32)(33,34)(35,44)(36,41)(37,42)(38,52)(39,43)(40,47)(45,60)(46,54)(48,57)(49,55)(50,58)(51,56)(53,62)(59,63)(61,64);(1,3,7,17)(2,8,12,24)(4,14,20,30)(5,16,6,15)(9,25,26,22)(10,23,11,13)(18,28,27,32)(19,31,21,29)(33,35,39,48)(34,40,43,54)(36,45,51,63)(37,47,38,46)(41,53,56,64)(42,57,52,44)(49,62,50,61)(55,59,58,60);(1,4,7,20)(2,9,12,26)(3,14,17,30)(5,19,6,21)(8,22,24,25)(10,18,11,27)(13,28,23,32)(15,29,16,31)(33,36,39,51)(34,41,43,56)(35,45,48,63)(37,50,38,49)(40,53,54,64)(42,58,52,55)(44,59,57,60)(46,62,47,61);(33,37,39,38)(34,42,43,52)(35,46,48,47)(36,49,51,50)(40,44,54,57)(41,55,56,58)(45,61,63,62)(53,60,64,59)

Polycyclic presentation:

2^9|b^2=h,c^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=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=di,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, 142709);