#494167

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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)(5,6,10,16,12,17,11,32)(7,18,22,43,24,44,23,52)(8,14,27,39,25,37,28,56)(9,29,30,40,31,57,19,15)(13,33,35,51,36,45,42,21)(20,41,49,64,47,62,50,67)(26,54,55,61,53,68,46,38)(34,58,60,66,59,65,63,48);(5,7,19,45,11,23,31,51,12,24,30,33,10,22,9,21)(6,13,29,43,16,35,40,44,17,36,57,52,32,42,15,18)(8,20,46,65,28,50,53,66,25,47,55,58,27,49,26,48)(14,34,54,64,39,60,61,62,37,59,68,67,56,63,38,41);(1,2,3,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,25)(13,34)(15,38)(16,39)(17,37)(18,41)(19,46)(21,48)(22,49)(23,50)(24,47)(29,54)(30,55)(31,53)(32,56)(33,58)(35,60)(36,59)(40,61)(42,63)(43,64)(44,62)(45,65)(51,66)(52,67)(57,68);(8,25)(14,37)(20,47)(26,53)(27,28)(34,59)(38,61)(39,56)(41,62)(46,55)(48,66)(49,50)(54,68)(58,65)(60,63)(64,67)

Polycyclic presentation:

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

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, 401363);