#7674644

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4338 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,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, 7581840);