#147874

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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	#1786
prime radical	#60643	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1128 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)(7,18)(8,24,27,61)(9,29)(10,35)(11,17)(12,20)(13,26,41,25)(14,33)(15,45)(16,32)(19,38,54,66)(21,40)(22,46)(23,52)(28,44,37,47)(30,49)(31,42,62,50)(34,48)(36,43)(39,58)(51,53,60,63)(55,68,59,67)(56,65,57,64);(1,2)(3,4)(5,7)(6,12)(8,19)(9,30)(10,36)(11,23)(13,38)(14,43)(15,46)(16,49)(17,40)(18,29)(20,33)(21,45)(22,32)(24,60)(25,57)(26,56)(27,54)(28,59)(31,63)(34,58)(35,39)(37,55)(41,66)(42,68)(44,65)(47,64)(48,52)(50,67)(51,61)(53,62);(2,4)(5,8)(6,13)(7,19)(9,31)(10,37)(11,27)(12,38)(14,44)(15,47)(16,50)(17,41)(18,51)(20,56)(21,57)(22,59)(23,54)(24,48)(25,45)(26,33)(28,32)(29,61)(30,63)(34,62)(35,42)(36,55)(39,68)(40,66)(43,65)(46,64)(49,67)(52,60)(53,58)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,h^b=hi,d^c=dhi,e^c=eh,g^c=gi,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, 55070);