#146553

order	512
nilpotency class	4
isoclinism	isoclinic class of #12933
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1853
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

965 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280192, #72280321, #72281021, …
maximal normal subgroup of	#72280192, #72280321, #72281021, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53749);