#149461

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

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

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, 56657);