#146773

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

946 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)(7,19)(8,24)(9,29)(10,34)(11,38)(12,17)(13,21)(14,26)(15,33)(16,37)(18,32)(20,41)(22,47)(23,54)(25,45)(27,48)(28,44)(30,43)(31,46)(35,51)(36,52)(39,49)(40,50)(42,59)(53,55)(56,66)(57,64)(58,63)(60,65)(61,68)(62,67);(5,7)(6,13,17,42)(8,20)(9,30)(10,35)(11,39)(12,23)(14,41,45,64)(15,47,16,49)(18,51,38,43)(19,29,54,34)(21,37)(22,32)(24,53,44,67)(25,61)(26,58)(27,60)(28,57)(31,62)(33,59)(36,55)(40,56)(46,66,52,65)(48,68,50,63);(5,8,12,28)(6,14,17,45)(7,20,23,57)(9,31,10,36)(11,40,32,27)(13,41,42,64)(15,48,16,50)(18,52,38,46)(19,53,54,67)(21,58,59,61)(22,60,39,56)(24,29,44,34)(25,37,26,33)(30,62,35,55)(43,66,51,65)(47,68,49,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=eh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282014, #72282032, #72282210
maximal normal subgroup of	#72282014, #72282032, #72282210

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53969);