#147705

order	512
nilpotency class	4
isoclinism	isoclinic class of #137655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

998 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54901);