#503178

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1100 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)(7,21,25,51)(8,26,32,59)(9,18,35,48)(10,19,12,17)(11,16,34,49)(14,37,40,24)(15,41,46,30)(22,53,56,67)(23,57,58,39)(27,52,54,66)(28,45,33,44)(29,47,31,43)(36,50,63,68)(38,62,65,60)(42,61,64,55);(5,7,10,24,13,25,12,37)(6,14,17,21,20,40,19,51)(8,27,29,60,32,54,31,62)(9,23,11,36,35,58,34,63)(15,42,44,53,46,64,45,67)(16,39,18,50,49,57,48,68)(22,28,55,41,56,33,61,30)(26,38,43,52,59,65,47,66);(1,2,3,4)(5,8,9,33,12,31,34,41,13,32,35,28,10,29,11,30)(6,15,16,47,19,45,48,59,20,46,49,43,17,44,18,26)(7,22,36,60,24,55,58,54,25,56,63,62,37,61,23,27)(14,38,50,53,21,52,57,64,40,65,68,67,51,66,39,42);(5,9,12,34,13,35,10,11)(6,16,19,48,20,49,17,18)(7,23,37,63,25,58,24,36)(8,28,31,30,32,33,29,41)(14,39,51,68,40,57,21,50)(15,43,45,26,46,47,44,59)(22,54,61,60,56,27,55,62)(38,64,66,53,65,42,52,67)

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,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, 410374);