#459319

order	512
nilpotency class	3
isoclinism	isoclinic class of #453999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1117 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 366515);