#146311

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1726
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

564 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)(5,6,9,14,10,15,25,36,11,16,26,37,27,38,49,60)(7,12,18,29,20,31,42,53,19,30,41,52,43,54,62,66)(8,13,22,33,24,35,47,58,23,34,46,57,48,59,64,68)(17,28,39,50,21,32,44,55,40,51,61,65,45,56,63,67);(2,4)(5,7)(6,12)(8,21,23,45)(9,18)(10,20)(11,19)(13,32,34,56)(14,29)(15,31)(16,30)(17,24,40,48)(22,44,46,63)(25,42)(26,41)(27,43)(28,35,51,59)(33,55,57,67)(36,53)(37,52)(38,54)(39,47,61,64)(49,62)(50,58,65,68)(60,66);(5,8)(6,13)(7,17,19,40)(9,22)(10,24)(11,23)(12,28,30,51)(14,33)(15,35)(16,34)(18,39,41,61)(20,21,43,45)(25,47)(26,46)(27,48)(29,50,52,65)(31,32,54,56)(36,58)(37,57)(38,59)(42,44,62,63)(49,64)(53,55,66,67)(60,68)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,h^2=i,b^a=bd,c^b=ch,g^b=gi,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53507);