#147219

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1862
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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:

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

Polycyclic presentation:

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

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, 54415);