#7659031

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	#36032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4442 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)(10,13)(11,15)(17,18,23,21)(19,27,30,24)(20,25,31,32)(22,26,35,34)(28,39,41,36)(29,40,43,37)(33,38,44,45)(42,47,48,46);(1,2,5,6)(3,7,4,8)(9,10,12,13)(11,14,15,16)(20,31)(25,32)(28,41)(33,44)(36,39)(38,45)(42,48)(46,47);(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,21,27)(20,28,31,41)(22,29,35,43)(25,36,32,39)(26,37,34,40)(33,42,44,48)(38,46,45,47);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,28)(21,32)(22,33)(23,31)(24,36)(26,38)(27,39)(29,42)(30,41)(34,45)(35,44)(37,46)(40,47)(43,48);(1,4,5,3)(2,8,6,7)(17,21,23,18)(19,27,30,24)(20,32,31,25)(22,26,35,34)(28,39,41,36)(29,37,43,40)(33,38,44,45)(42,46,48,47);(17,22)(18,26)(19,29)(20,33)(21,34)(23,35)(24,37)(25,38)(27,40)(28,42)(30,43)(31,44)(32,45)(36,46)(39,47)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566227);