#148337

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11096
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

647 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55533);