#146542

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1824
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

950 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)(5,6,10,17,12,16,11,34)(7,18,22,47,21,48,35,43)(8,23,25,42,27,14,26,41)(9,28,31,45,33,60,32,15)(13,29,37,54,39,62,38,20)(19,46,51,68,50,64,52,67)(24,56,58,44,30,63,59,40)(36,57,65,49,55,61,66,53);(5,7,10,22,12,21,11,35)(6,13,17,37,16,39,34,38)(8,19,26,52,27,50,25,51)(9,29,31,54,33,62,32,20)(14,36,42,66,23,55,41,65)(15,43,28,18,45,47,60,48)(24,57,59,53,30,61,58,49)(40,67,63,64,44,68,56,46);(2,4)(5,8)(6,14)(7,19)(9,30)(10,26)(11,25)(12,27)(13,36)(15,44)(16,23)(17,42)(18,46)(20,53)(21,50)(22,52)(24,33)(28,56)(29,61)(31,58)(32,59)(34,41)(35,51)(37,66)(38,65)(39,55)(40,45)(43,68)(47,67)(48,64)(49,54)(57,62)(60,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,d^b=dh,e^b=ei,h^b=hi,d^c=di,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53738);