#7674634

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5210 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,e^2=i,f^2=i,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, 7581830);