#146324

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=di,g^a=gi,e^b=ei,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53520);