#7673612

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

4314 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=hi,e^2=gh,f^2=g,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, 7580808);