#7673664

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

4325 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gi,e^2=g,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, 7580860);