#454001

order	512
nilpotency class	3
isoclinism	isoclinic class of #453999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2129 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,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, 361197);