#494437

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60450	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1374 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)(5,6,10,16,11,17,31,44,12,18,32,45,30,43,9,15)(7,19,22,37,23,42,21,46,24,48,29,13,33,35,53,36)(8,14,27,40,28,41,54,62,25,38,55,63,56,64,26,39)(20,47,51,59,52,65,50,66,49,67,57,34,58,60,68,61);(5,7,11,33,12,24,30,23)(6,13,17,46,18,37,43,36)(8,20,28,58,25,49,56,52)(9,29,10,21,31,22,32,53)(14,34,41,66,38,59,64,61)(15,42,16,19,44,35,45,48)(26,57,27,50,54,51,55,68)(39,65,40,47,62,60,63,67);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,25)(13,34)(15,39)(16,40)(17,41)(18,38)(19,47)(21,50)(22,51)(23,52)(24,49)(29,57)(30,56)(31,54)(32,55)(33,58)(35,60)(36,61)(37,59)(42,65)(43,64)(44,62)(45,63)(46,66)(48,67)(53,68);(8,25)(14,38)(20,49)(26,54)(27,55)(28,56)(34,59)(39,62)(40,63)(41,64)(47,67)(50,68)(51,57)(52,58)(60,65)(61,66)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323138
maximal normal subgroup of	#72323138
maximal quotient of	#72323090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401633);