#122140

order	512
nilpotency class	3
isoclinism	isoclinic class of #122137
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=eh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29336);