#7715463

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	#39335	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4860 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622659);