#111988

order	512
nilpotency class	3
isoclinism	isoclinic class of #111981
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 19184);