#146545

order	512
nilpotency class	4
isoclinism	isoclinic class of #126723
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1736
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10730
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

963 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280321, #72280507, #72281010, …
maximal normal subgroup of	#72280321, #72280507, #72281010, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53741);