#148861

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1804
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

488 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,e^b=ei,d^c=di,e^c=eh,f^c=fi,g^c=gh,h^c=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, 56057);