#147095

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1846
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=h,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,f^b=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, 54291);