#249871

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	#61800	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

937 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,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, 157067);