#347455

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
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	#63048	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2144 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)(12,29,33,43)(13,34,36,46)(14,21,15,20)(17,27,41,26)(18,31,44,32)(19,37,45,35)(24,49,52,38)(25,39,55,64)(30,42,60,66)(47,56,65,51)(48,53,61,54)(50,67,71,72)(57,59,62,58)(63,70,68,69);(3,7)(5,8)(9,11)(10,17)(12,24)(13,25)(14,27)(15,26)(16,28)(18,38)(19,39)(20,40)(21,23)(22,41)(29,47)(30,50)(31,56)(32,51)(33,52)(34,61)(35,54)(36,55)(37,53)(42,63)(43,65)(44,49)(45,64)(46,48)(57,67)(58,70)(59,69)(60,71)(62,72)(66,68);(1,2)(3,8)(4,6)(5,7)(9,12)(10,18)(11,24)(13,30)(14,31)(15,32)(16,33)(17,38)(19,42)(20,43)(21,29)(22,44)(23,47)(25,50)(26,51)(27,56)(28,52)(34,62)(35,58)(36,60)(37,59)(39,63)(40,65)(41,49)(45,66)(46,57)(48,67)(53,69)(54,70)(55,71)(61,72)(64,68);(9,13,16,36)(10,19,22,45)(11,25,28,55)(12,30,33,60)(14,35,15,37)(17,39,41,64)(18,42,44,66)(20,34,21,46)(23,48,40,61)(24,50,52,71)(26,53,27,54)(29,57,43,62)(31,58,32,59)(38,63,49,68)(47,67,65,72)(51,69,56,70)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209895, #72209899, #72268466, …
maximal normal subgroup of	#72209895, #72268466
maximal quotient of	#72227833, #72228007, #72228031

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254651);