#417664

order	512
nilpotency class	3
isoclinism	isoclinic class of #417661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

2325 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:

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

Polycyclic presentation:

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

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, 324860);