#149072

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2087
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

448 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56268);