#459302

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
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

1095 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:

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

Polycyclic presentation:

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

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, 366498);