#10320437

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(4,8)(6,7)(9,10,14,12)(11,13,16,15)(17,18)(19,23)(20,27)(21,25)(22,29)(24,32)(26,35)(28,33)(30,39)(31,37)(34,41)(36,42)(38,45)(40,44)(43,46)(47,48);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,13)(14,16)(18,19)(21,33)(24,27)(26,29)(30,42)(34,45)(37,40)(43,48);(1,2,5,3)(4,6,8,7)(9,10,14,12)(11,15,16,13)(17,19,23,18)(20,24,32,27)(21,28,33,25)(22,29,35,26)(30,36,42,39)(31,37,44,40)(34,41,45,38)(43,46,48,47);(2,3)(6,7)(10,12)(11,16)(17,20,23,32)(18,24,19,27)(21,30,33,42)(22,31,35,44)(25,36,28,39)(26,37,29,40)(34,43,45,48)(38,46,41,47);(1,3,5,2)(4,6,8,7)(9,12,14,10)(11,15,16,13)(17,21)(18,25)(19,28)(20,30)(22,34)(23,33)(24,36)(26,38)(27,39)(29,41)(31,43)(32,42)(35,45)(37,46)(40,47)(44,48);(1,4,5,8)(2,6,3,7)(9,13,14,15)(10,11,12,16)(17,22)(18,26)(19,29)(20,31)(21,34)(23,35)(24,37)(25,38)(27,40)(28,41)(30,43)(32,44)(33,45)(36,46)(39,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268513
Fitting subgroup of	#72268513
maximal subgroup of	#72268513
maximal normal subgroup of	#72268513
maximal quotient of	#72268481

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227633);