#363506

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1240 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)(7,18)(8,14)(9,29)(10,33)(11,16)(12,17)(13,21)(15,30)(19,37)(20,49)(22,45)(23,36)(24,53)(25,55)(26,42)(27,39)(28,40)(31,60)(32,41)(34,47)(35,52)(38,56)(43,48)(44,50)(46,66)(51,65)(54,64)(57,59)(58,63)(61,68)(62,67);(5,7,10,21,12,22,11,23)(6,13,16,18,17,36,33,45)(8,24,26,19,28,47,27,48)(9,20,30,50,31,52,32,51)(14,37,39,53,40,43,42,34)(15,35,29,44,41,49,60,65)(25,54,56,46,57,67,58,68)(38,62,55,66,63,64,59,61);(5,8,12,28)(6,14,17,40)(7,19,22,48)(9,25,31,57)(10,27,11,26)(13,34,36,53)(15,38,41,63)(16,42,33,39)(18,43,45,37)(20,46,52,68)(21,24,23,47)(29,59,60,55)(30,58,32,56)(35,61,49,66)(44,64,65,62)(50,54,51,67);(5,9,12,31)(6,15)(7,20,22,52)(8,25,28,57)(10,30,11,32)(13,35)(14,38)(16,29)(17,41)(18,44)(19,46,48,68)(21,50,23,51)(24,54,47,67)(26,56,27,58)(33,60)(34,61)(36,49)(37,62)(39,55)(40,63)(42,59)(43,64)(45,65)(53,66)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270702);