#148331

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2076
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

658 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:

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

Polycyclic presentation:

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

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, 55527);