#423244

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1468 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)(7,19,23,37)(8,14,29,42)(9,15,33,46)(10,18,12,16)(13,24,36,22)(20,38,50,62)(21,47,52,43)(25,34,49,63)(26,54,57,39)(27,41,28,40)(30,59,51,35)(31,45,32,44)(48,68,55,64)(53,65,60,61)(56,66,58,67);(1,2)(3,4)(5,7,12,24,11,23,10,22)(6,13,18,37,17,36,16,19)(8,25,28,50,29,49,27,20)(9,30,32,21,33,51,31,52)(14,38,41,63,42,62,40,34)(15,43,45,35,46,47,44,59)(26,55,58,53,57,48,56,60)(39,65,67,64,54,61,66,68);(2,4)(5,8,12,27,11,29,10,28)(6,14,18,40,17,42,16,41)(7,20,24,49,23,50,22,25)(9,26,31,58,33,57,32,56)(13,34,37,62,36,63,19,38)(15,39,44,67,46,54,45,66)(21,48,30,60,52,55,51,53)(35,61,43,68,59,65,47,64);(5,9)(6,15)(7,21)(8,26)(10,32)(11,33)(12,31)(13,35)(14,39)(16,45)(17,46)(18,44)(19,47)(20,48)(22,51)(23,52)(24,30)(25,53)(27,58)(28,56)(29,57)(34,61)(36,59)(37,43)(38,64)(40,67)(41,66)(42,54)(49,60)(50,55)(62,68)(63,65)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330440);