#7630930

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

4794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538126);