#494473

order	512
nilpotency class	4
isoclinism	isoclinic class of #481319
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
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 ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1356 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:

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

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,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343344, #72343359, #72343362, …
maximal normal subgroup of	#72343344, #72343359, #72343362, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401669);