#249867

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	#61944	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

931 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)(3,11,13,25)(4,8,15,23)(5,17,18,9)(7,19,22,14)(12,28,29,27)(16,26,20,24)(21,31,32,30)(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);(4,15)(5,18)(8,23)(9,17)(11,25)(12,29)(14,19)(21,32)(33,35)(34,40)(36,50)(37,48)(38,45)(39,47)(41,58)(42,55)(43,57)(44,53)(46,52)(49,64)(51,63)(54,62)(56,59)(60,61);(1,3)(2,7)(4,12)(5,19)(6,13)(8,21)(9,25)(10,22)(11,17)(14,18)(15,29)(16,31)(20,30)(23,32)(24,27)(26,28)(35,45)(37,38)(40,55)(42,53)(44,57)(46,62)(47,48)(49,60)(50,63)(51,54)(56,64)(58,61);(1,4,6,15)(2,8,10,23)(3,12,13,29)(5,20,18,16)(7,21,22,32)(9,26,17,24)(11,27,25,28)(14,31,19,30)(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,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, 157063);