#451052

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1238 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)(7,19)(8,25,30,46)(9,31)(10,17)(11,16)(12,18)(13,22)(14,29,42,28)(15,33)(20,47,50,36)(21,51)(23,39)(24,38)(26,40,49,63)(27,56,60,41)(32,43)(34,45)(35,44)(37,53)(48,67,57,64)(52,62)(54,59,66,58)(55,65,68,61);(5,7,12,24)(6,13,18,39)(8,26,30,49)(9,32,35,52)(10,23,11,22)(14,40,42,63)(15,43,45,62)(16,38,17,19)(20,29,50,28)(21,34,53,33)(25,36,46,47)(27,57,60,48)(31,37,44,51)(41,65,56,61)(54,64,66,67)(55,59,68,58);(5,8)(6,14)(7,20)(9,27)(10,28)(11,29)(12,30)(13,36)(15,41)(16,25)(17,46)(18,42)(19,40)(21,48)(22,26)(23,49)(24,50)(31,54)(32,55)(33,58)(34,59)(35,60)(37,61)(38,63)(39,47)(43,64)(44,66)(45,56)(51,65)(52,68)(53,57)(62,67);(2,4)(5,9,11,34,12,35,10,33)(6,15,17,44,18,45,16,31)(7,21,23,52,24,53,22,32)(8,27,29,59,30,60,28,58)(13,37,38,62,39,51,19,43)(14,41,46,66,42,56,25,54)(20,48,49,68,50,57,26,55)(36,61,63,67,47,65,40,64)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=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, 358248);