#187732

order	512
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62743	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2050 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)(3,12,15,8)(4,16)(5,10)(6,19)(7,11)(9,17)(13,28,30,32)(14,24,20,23)(18,25)(21,26)(22,31,27,29)(33,34,38,43,39,44,37,42)(35,45,49,40,48,55,47,54)(36,41,50,56,51,57,52,58)(46,60,61,64,62,59,63,53);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,22)(10,23)(11,12)(16,28)(17,29)(18,30)(19,24)(21,31)(25,32)(26,27)(33,35)(34,40)(36,46)(37,49)(38,47)(39,48)(41,53)(42,55)(43,45)(44,54)(50,63)(51,62)(52,61)(56,59)(57,64)(58,60);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,26)(12,27)(14,29)(15,30)(16,19)(20,31)(23,32)(24,28)(33,36)(34,41)(35,46)(37,52)(38,50)(39,51)(40,53)(42,58)(43,56)(44,57)(45,59)(47,63)(48,62)(49,61)(54,64)(55,60);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,17,18,21)(8,23,12,24)(9,25,26,16)(13,29,30,31)(22,32,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94928);