#347430

order	512
nilpotency class	3
isoclinism	isoclinic class of #347034
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62975	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1976 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,f^c=fi,g^c=gh,e^d=ehi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254626);