#151750

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₄²
commutator	C₄	C₄²×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

276 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,g^2=h,h^2=i,b^a=bh,e^a=ei,c^b=ci,d^b=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210687, #72234252
maximal normal subgroup of	#72210687, #72234252
maximal quotient of	#72210680, #72234239

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58946);