#244611

order	512
nilpotency class	3
isoclinism	isoclinic class of #244583
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	#3207	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

944 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)(3,12)(4,9)(5,17)(6,19)(7,11)(8,13)(10,16)(14,22)(15,24)(18,25)(20,23)(21,26)(27,29)(28,31)(30,32)(33,34,39,44)(35,40,47,56)(36,48,50,59)(37,43,38,42)(41,52,58,51)(45,57,61,64)(46,60,53,55)(49,54,62,63);(1,3,7,15)(2,8,11,23)(4,13,6,20)(5,14,18,30)(9,24,19,12)(10,22,26,32)(16,28,21,29)(17,27,25,31)(33,35)(34,40)(36,49)(37,46)(38,53)(39,47)(41,57)(42,55)(43,60)(44,56)(45,52)(48,63)(50,62)(51,61)(54,59)(58,64);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,16,18,21)(8,12,23,24)(10,25,26,17)(14,28,30,29)(22,27,32,31)(33,36)(34,41)(35,45)(37,52)(38,51)(39,50)(40,54)(42,59)(43,48)(44,58)(46,62)(47,61)(49,53)(55,64)(56,63)(57,60);(1,5,7,18)(2,10,11,26)(3,14,15,30)(4,16,6,21)(8,22,23,32)(9,25,19,17)(12,27,24,31)(13,28,20,29)(33,35)(34,40)(36,45)(37,53)(38,46)(39,47)(41,54)(42,60)(43,55)(44,56)(48,64)(49,52)(50,61)(51,62)(57,59)(58,63)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151807);