#247042

order	512
nilpotency class	3
isoclinism	isoclinic class of #247017
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	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

965 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,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, 154238);