#187743

order	512
nilpotency class	3
isoclinism	isoclinic class of #187721
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62828	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2028 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)(3,11,13,7)(4,15)(5,18)(6,9)(8,17)(10,21)(12,28,29,32)(14,23,19,25)(16,26)(20,24)(22,30,27,31)(33,34,38,44,39,42,37,43)(35,45,49,40,48,55,47,54)(36,41,50,57,51,56,52,58)(46,60,61,64,62,59,63,53);(1,3)(2,7)(4,12)(5,19)(6,13)(8,22)(9,11)(10,25)(14,21)(15,28)(16,29)(17,31)(18,23)(20,30)(24,27)(26,32)(33,35)(34,40)(36,46)(37,49)(38,47)(39,48)(41,53)(42,54)(43,55)(44,45)(50,63)(51,62)(52,61)(56,64)(57,59)(58,60);(1,4)(2,8)(3,12)(5,20)(6,16)(7,22)(9,24)(10,26)(11,27)(13,29)(14,31)(15,18)(17,21)(19,30)(23,28)(25,32)(33,36)(34,41)(35,46)(37,52)(38,50)(39,51)(40,53)(42,56)(43,58)(44,57)(45,59)(47,63)(48,62)(49,61)(54,64)(55,60);(2,9)(3,13)(8,24)(10,18)(12,29)(14,19)(15,26)(30,31)(34,42)(40,54)(41,56)(43,44)(45,55)(53,64)(57,58)(59,60)

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94939);