#503176

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
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×#3290	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1105 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)(5,6,12,18)(7,19,24,41)(8,25,30,61)(9,17,33,50)(10,34,35,16)(11,36,32,15)(13,23,42,22)(14,43,47,29)(20,53,57,68)(21,40,59,39)(26,51,54,65)(27,46,31,60)(28,48,64,45)(37,52,58,49)(38,63,66,62)(44,56,67,55);(5,7,10,22,12,24,35,23)(6,13,16,19,18,42,34,41)(8,26,28,62,30,54,64,63)(9,21,32,58,33,59,11,37)(14,44,46,53,47,67,60,68)(15,39,17,49,36,40,50,52)(20,31,55,43,57,27,56,29)(25,38,45,51,61,66,48,65);(1,2,3,4)(5,8,9,31,10,28,32,43,12,30,33,27,35,64,11,29)(6,14,36,45,16,46,50,61,18,47,15,48,34,60,17,25)(7,20,37,63,23,56,59,54,24,57,58,62,22,55,21,26)(13,38,49,68,41,65,39,67,42,66,52,53,19,51,40,44);(5,9,10,32,12,33,35,11)(6,15,16,17,18,36,34,50)(7,21,22,58,24,59,23,37)(8,27,28,29,30,31,64,43)(13,39,19,49,42,40,41,52)(14,45,46,61,47,48,60,25)(20,54,55,63,57,26,56,62)(38,44,51,53,66,67,65,68)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=e,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,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410372);