#494486

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei,g^c=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, 401682);