#237524

order	512
nilpotency class	3
isoclinism	isoclinic class of #237505
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	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1113 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)(3,13,15,25)(4,17,19,26)(5,11,6,10)(8,20,23,16)(9,21,24,18)(14,22,28,32)(27,29,31,30)(33,34)(35,40)(36,48)(37,42)(38,43)(39,44)(41,50)(45,58)(46,55)(47,57)(49,54)(51,59)(52,60)(53,56)(61,64)(62,63);(1,3,7,15)(2,8,12,23)(4,14,19,28)(5,20,6,16)(9,22,24,32)(10,25,11,13)(17,31,26,27)(18,30,21,29)(33,35)(34,40)(36,49)(37,46)(38,53)(39,47)(41,58)(42,55)(43,56)(44,57)(45,52)(48,63)(50,61)(51,62)(54,59)(60,64);(1,4,7,19)(2,9,12,24)(3,14,15,28)(5,18,6,21)(8,22,23,32)(10,17,11,26)(13,27,25,31)(16,29,20,30)(33,36)(34,41)(35,45)(37,52)(38,50)(39,51)(40,54)(42,48)(43,59)(44,60)(46,62)(47,61)(49,53)(55,58)(56,64)(57,63);(3,15)(8,23)(13,25)(14,28)(16,20)(22,32)(27,31)(29,30)(33,37,39,38)(34,42,44,43)(35,46,47,53)(36,50,51,52)(40,55,57,56)(41,59,60,48)(45,49,61,62)(54,64,63,58)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144720);