#7673591

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

4311 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=i,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, 7580787);