#497854

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

598 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)(7,21)(8,26)(9,16)(10,19)(11,37)(12,17)(13,20)(14,24)(15,30)(18,23)(22,52)(25,49)(27,47)(28,51)(29,58)(31,57)(32,59)(33,50)(34,42)(35,41)(36,48)(38,40)(39,61)(43,62)(44,53)(45,60)(46,65)(54,56)(55,67)(63,68)(64,66);(5,7)(6,14)(8,22)(9,11)(10,34)(12,24)(13,25)(15,39)(16,47)(17,49)(18,41)(19,21)(20,42)(23,33)(26,56)(27,36)(28,30)(29,62)(31,54)(32,55)(35,38)(37,50)(40,48)(43,59)(44,68)(45,51)(46,66)(52,58)(53,61)(57,67)(60,64)(63,65);(5,8,27,60,12,31,11,30,13,32,23,53,10,29,38,65)(6,15,40,57,19,45,18,26,20,46,37,58,17,44,47,59)(7,22,36,64,24,54,9,28,25,55,33,61,34,62,35,63)(14,39,48,67,21,51,41,56,42,66,50,52,49,68,16,43);(5,9)(6,16)(7,23)(8,28)(10,36)(11,34)(12,33)(13,35)(14,40)(15,43)(17,50)(18,21)(19,48)(20,41)(22,53)(24,38)(25,27)(26,51)(29,64)(30,62)(31,61)(32,63)(37,42)(39,57)(44,52)(45,67)(46,56)(47,49)(54,65)(55,60)(58,66)(59,68)

Polycyclic presentation:

2^9|a^2=g,c^2=fhi,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405050);