#8115473

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	#34643	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4333 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022669);