#187796

order	512
nilpotency class	3
isoclinism	isoclinic class of #187780
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1966 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)(3,11,14,7)(4,8)(5,9)(6,10)(12,27,29,20)(13,24,17,21)(15,22)(16,23)(18,25)(19,26)(28,32,30,31)(33,34)(35,43,46,55)(36,47,49,40)(37,41)(38,42)(39,45,53,50)(44,58)(48,62,64,54)(51,56)(52,59)(57,60)(61,63);(1,3)(2,7)(4,12)(5,17)(6,14)(8,20)(9,24)(10,11)(13,19)(15,30)(16,29)(18,28)(21,26)(22,32)(23,27)(25,31)(33,35,38,46)(34,39,42,53)(36,44,49,60)(37,50,51,45)(40,52,47,61)(41,55,56,43)(48,63,64,59)(54,58,62,57);(1,4,6,16)(2,8,10,23)(3,12,14,29)(5,18,19,15)(7,20,11,27)(9,25,26,22)(13,30,17,28)(21,32,24,31)(33,36)(34,40)(35,44)(37,48)(38,49)(39,52)(41,54)(42,47)(43,57)(45,59)(46,60)(50,63)(51,64)(53,61)(55,58)(56,62);(1,5,6,19)(2,9,10,26)(3,13,14,17)(4,15,16,18)(7,21,11,24)(8,22,23,25)(12,28,29,30)(20,31,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=ghi,d^c=dh,g^c=gi

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, 94992);