#249875

order	512
nilpotency class	3
isoclinism	isoclinic class of #249839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3115	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

907 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh,e^d=eh,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, 157071);