#348658

order	512
nilpotency class	3
isoclinism	isoclinic class of #348654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3529	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

845 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,d^b=dh,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255854);