#149000

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1852
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

474 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:

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

Polycyclic presentation:

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

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, 56196);