#503172

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3293	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1094 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)(5,6,13,20)(7,21,25,49)(8,26,32,58)(9,18,34,46)(10,19,12,17)(11,16,33,47)(14,36,39,24)(15,40,45,30)(22,51,54,67)(23,55,56,38)(27,50,61,65)(28,43,52,44)(29,42,31,57)(35,48,63,68)(37,62,64,60)(41,59,66,53);(5,7,10,24,13,25,12,36)(6,14,17,21,20,39,19,49)(8,27,29,60,32,61,31,62)(9,23,11,35,34,56,33,63)(15,41,43,51,45,66,44,67)(16,38,18,48,47,55,46,68)(22,52,53,40,54,28,59,30)(26,37,57,50,58,64,42,65);(1,2,3,4)(5,8,9,28,12,31,33,40,13,32,34,52,10,29,11,30)(6,15,16,42,19,44,46,58,20,45,47,57,17,43,18,26)(7,22,35,60,24,53,56,61,25,54,63,62,36,59,23,27)(14,37,48,51,21,50,55,66,39,64,68,67,49,65,38,41);(5,9,12,33,13,34,10,11)(6,16,19,46,20,47,17,18)(7,23,36,63,25,56,24,35)(8,28,31,40,32,52,29,30)(14,38,49,68,39,55,21,48)(15,42,44,58,45,57,43,26)(22,27,59,62,54,61,53,60)(37,41,65,67,64,66,50,51)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410368);