#220184

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,3,5,4,6,7,8)(9,10)(11,21)(12,26,29,17)(13,18)(14,33)(15,20)(16,24)(19,36)(22,49,52,47)(23,48)(25,45)(27,56,60,40)(28,41,63,57)(30,64)(31,46)(32,44)(34,39)(35,43)(37,51,55,61)(38,54)(42,53)(50,66,70,69)(58,65,71,72)(59,67,62,68);(9,11,15,25)(10,16,20,39)(12,22,29,52)(13,30,32,54)(14,34,36,24)(17,37,26,55)(18,42,44,48)(19,45,33,21)(23,35,53,31)(27,58,60,71)(28,61,63,51)(38,46,64,43)(40,66,56,69)(41,49,57,47)(50,62,70,59)(65,68,72,67);(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,37)(18,40)(19,41)(20,26)(21,47)(23,50)(24,51)(25,52)(30,58)(31,59)(32,60)(33,57)(34,61)(35,62)(36,63)(38,65)(39,55)(42,66)(43,67)(44,56)(45,49)(46,68)(48,69)(53,70)(54,71)(64,72);(9,13,15,32)(10,18,20,44)(11,23,25,53)(12,27,29,60)(14,35,36,31)(16,38,39,64)(17,40,26,56)(19,46,33,43)(21,48,45,42)(22,50,52,70)(24,54,34,30)(28,62,63,59)(37,65,55,72)(41,68,57,67)(47,69,49,66)(51,71,61,58)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,g^c=gh,e^d=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127380);