#455965

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
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

1098 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:

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

Polycyclic presentation:

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

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, 363161);