#493827

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#466
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

944 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)(5,6,9,15,12,17,11,33)(7,18,21,43,24,45,23,54)(8,25,28,40,29,38,26,14)(10,31,30,41,32,60,19,16)(13,34,36,53,37,46,44,22)(20,48,51,66,52,65,49,42)(27,39,47,56,58,64,57,59)(35,50,55,67,63,68,62,61);(5,7,19,46,11,23,32,53,12,24,30,34,9,21,10,22)(6,13,31,43,15,36,41,45,17,37,60,54,33,44,16,18)(8,20,47,67,28,51,58,68,29,52,57,61,26,49,27,50)(14,35,59,65,38,62,64,66,40,63,56,48,25,55,39,42);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,29)(13,35)(15,38)(16,39)(17,40)(18,42)(19,47)(21,49)(22,50)(23,51)(24,52)(25,33)(30,57)(31,59)(32,58)(34,61)(36,62)(37,63)(41,64)(43,65)(44,55)(45,66)(46,67)(48,54)(53,68)(56,60);(5,9,12,11)(6,15,17,33)(7,21,24,23)(8,26,29,28)(10,30,32,19)(13,36,37,44)(14,38,40,25)(16,31,41,60)(18,43,45,54)(20,49,52,51)(22,34,53,46)(27,57,58,47)(35,62,63,55)(39,59,64,56)(42,65,66,48)(50,61,68,67)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401023);