#477004

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1016 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384200);