#249866

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	#535	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

914 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)(4,17)(5,14)(6,25)(7,10)(8,11)(9,21)(13,30)(15,24)(16,23)(18,29)(19,31)(20,28)(22,27)(26,32)(33,34,38,42)(35,43,45,57)(36,40,47,55)(37,49,50,41)(39,51,54,46)(44,60,61,59)(48,58,52,56)(53,63,64,62);(1,3)(2,5)(4,18)(6,15)(7,14)(8,16)(9,26)(10,23)(11,24)(12,25)(13,20)(17,19)(21,31)(22,32)(27,30)(28,29)(36,47)(37,50)(40,55)(41,49)(43,57)(44,61)(46,51)(53,64);(1,4,8,22)(2,9,11,28)(3,13,16,31)(5,19,24,30)(6,20,7,21)(10,27,25,17)(12,29,23,26)(14,18,15,32)(33,35)(34,39)(36,44)(37,51)(38,45)(40,53)(41,57)(42,54)(43,49)(46,50)(47,61)(48,63)(52,62)(55,64)(56,59)(58,60);(1,5)(2,3)(4,19)(6,23)(7,12)(8,24)(9,13)(10,15)(11,16)(14,25)(17,18)(20,26)(21,29)(22,30)(27,32)(28,31)(33,36,38,47)(34,40,42,55)(35,44,45,61)(37,52,50,48)(39,53,54,64)(41,58,49,56)(43,59,57,60)(46,63,51,62)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,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, 157062);