#7673667

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	#44758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4820 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)(3,7)(4,8)(5,6)(10,14)(15,16)(21,34)(25,38)(29,43)(32,45)(36,40)(37,44)(41,48)(46,47);(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,27,30,23)(20,24,33,31)(21,25,34,38)(28,39,42,35)(29,40,43,36)(32,37,45,44)(41,47,48,46);(2,6)(7,8)(17,18)(19,23)(20,31)(21,25)(22,26)(24,33)(27,30)(28,39)(29,36)(32,44)(34,38)(35,42)(37,45)(40,43)(41,47)(46,48);(1,2)(3,7)(4,8)(5,6)(17,19,22,30)(18,23,26,27)(20,28,33,42)(21,29,34,43)(24,35,31,39)(25,36,38,40)(32,41,45,48)(37,46,44,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,28)(21,32)(22,33)(23,35)(25,37)(26,31)(27,39)(29,41)(30,42)(34,45)(36,46)(38,44)(40,47)(43,48);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,16,14,15)(17,21)(18,25)(19,29)(20,32)(22,34)(23,36)(24,37)(26,38)(27,40)(28,41)(30,43)(31,44)(33,45)(35,46)(39,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580863);