#498577

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405773);