#244646

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	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

948 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:

(1,2,6,10)(3,11,14,25)(4,15,17,26)(5,18,21,9)(7,19,22,13)(8,20,24,16)(12,23,29,31)(27,30,32,28)(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);(1,3,6,14)(2,7,10,22)(4,12,17,29)(5,19,21,13)(8,23,24,31)(9,25,18,11)(15,27,26,32)(16,30,20,28)(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);(7,22)(11,25)(23,31)(27,32)(33,36,39,50)(34,41,44,58)(35,45,47,61)(37,52,38,51)(40,54,56,63)(42,59,43,48)(46,62,53,49)(55,64,60,57);(1,4,6,17)(2,8,10,24)(3,12,14,29)(5,20,21,16)(7,23,22,31)(9,26,18,15)(11,27,25,32)(13,30,19,28)(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=hi,b^2=h,c^2=i,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, 151842);