#148356

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
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

649 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:

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

Polycyclic presentation:

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

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, 55552);