#7673658

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

4333 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)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,24,27,28)(21,25,35,31)(22,32,36,26)(29,37,33,39)(30,42,41,38)(34,44,46,40)(43,48,45,47);(1,3,6,5)(2,7,8,4)(9,11,13,16)(10,14,12,15)(18,20)(19,27)(25,31)(26,32)(29,33)(30,41)(40,44)(43,45);(2,8)(4,7)(17,19)(18,24)(20,28)(21,33)(22,30)(23,27)(25,39)(26,38)(29,35)(31,37)(32,42)(34,45)(36,41)(40,48)(43,46)(44,47);(1,4)(2,3)(5,8)(6,7)(17,20,23,18)(19,28,27,24)(21,31,35,25)(22,32,36,26)(29,39,33,37)(30,42,41,38)(34,44,46,40)(43,48,45,47);(9,12,13,10)(11,15,16,14)(17,21,23,35)(18,25,20,31)(19,29,27,33)(22,34,36,46)(24,37,28,39)(26,40,32,44)(30,43,41,45)(38,47,42,48);(1,5,6,3)(2,4,8,7)(17,22)(18,26)(19,30)(20,32)(21,34)(23,36)(24,38)(25,40)(27,41)(28,42)(29,43)(31,44)(33,45)(35,46)(37,47)(39,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=i,e^2=g,f^2=h,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, 7580854);