#495110

order	512
nilpotency class	3
isoclinism	isoclinic class of #23414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1955
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1105 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239242, #72249832, #72249836, …
maximal normal subgroup of	#72239242, #72249832, #72249836, …
maximal quotient of	#72239260, #72322952

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402306);