#147001

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

512 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)(7,20)(8,26,29,42)(9,17)(10,16)(11,15)(12,19)(13,22)(14,28,40,58)(18,31)(21,48,51,62)(23,46)(24,35)(25,47)(27,39,59,64)(30,57,60,41)(32,43)(33,44)(34,52,61,68)(36,53)(37,55)(38,54)(45,50,65,66)(49,56,67,63);(2,4)(5,7,11,24,10,23,32,54,12,25,33,55,31,53,9,22)(6,13,17,36,18,37,44,47,19,38,43,46,16,35,15,20)(8,21,30,52,28,50,59,67,29,51,60,68,58,66,27,49)(14,34,41,62,42,63,64,65,40,61,57,48,26,56,39,45);(5,8,12,29)(6,14)(7,21,25,51)(9,27,32,59)(10,28,31,58)(11,30,33,60)(13,34)(15,39)(16,26)(17,41)(18,42)(19,40)(20,45)(22,49,54,67)(23,50,53,66)(24,52,55,68)(35,56)(36,62)(37,63)(38,61)(43,57)(44,64)(46,48)(47,65)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,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, 54197);