#495108

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1955
prime radical	C₂×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1089 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265582, #72265589, #72265606
maximal normal subgroup of	#72265582, #72265589, #72265606

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402304);