#354856

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#381
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	C₄×#381
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1369 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317342, #72317365
maximal normal subgroup of	#72317342, #72317365

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262052);