#347000

order	512
nilpotency class	3
isoclinism	isoclinic class of #346950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2109 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gi,e^c=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254196);