#7630827

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538023);