#147969

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6,10,16,11,17,30,43,12,18,31,44,9,15,29,42)(7,13,22,35,23,36,51,61,20,33,48,59,21,34,50,60)(8,14,27,40,26,39,55,65,25,38,54,64,28,41,56,66)(19,45,47,67,24,52,53,68,46,32,57,58,49,37,62,63);(1,2,3,4)(5,7)(6,13)(8,24,25,49)(9,21)(10,22)(11,23)(12,20)(14,37,38,52)(15,34)(16,35)(17,36)(18,33)(19,26,46,28)(27,53,54,62)(29,50)(30,51)(31,48)(32,39,45,41)(40,63,64,68)(42,60)(43,61)(44,59)(47,55,57,56)(58,65,67,66);(5,8)(6,14)(7,19)(9,28)(10,27)(11,26)(12,25)(13,32)(15,41)(16,40)(17,39)(18,38)(20,46)(21,49)(22,47)(23,24)(29,56)(30,55)(31,54)(33,45)(34,52)(35,58)(36,37)(42,66)(43,65)(44,64)(48,57)(50,62)(51,53)(59,67)(60,68)(61,63)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55165);