#7715443

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4478 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)(10,13)(14,16)(17,18)(19,28)(20,21)(22,38)(23,24)(25,29)(26,37)(27,39)(30,35)(31,33)(32,40)(34,47)(36,41)(42,43)(44,45)(46,48);(1,3)(2,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(18,24)(19,29)(20,26)(27,38)(31,35)(32,43)(34,41)(45,48);(2,4)(6,7)(17,19)(18,25)(20,33)(21,35)(22,32)(23,29)(24,28)(26,30)(27,40)(31,37)(34,46)(36,48)(38,42)(39,43)(41,44)(45,47);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,23,26)(18,21,24,37)(19,30,29,33)(22,34,39,41)(25,31,28,35)(27,36,38,47)(32,44,43,46)(40,45,42,48);(1,3)(2,6)(4,7)(5,8)(17,21)(18,26)(19,31)(20,24)(22,36)(23,37)(25,33)(27,41)(28,30)(29,35)(32,45)(34,38)(39,47)(40,46)(42,44)(43,48);(17,22,23,39)(18,27,24,38)(19,32,29,43)(20,34,26,41)(21,36,37,47)(25,40,28,42)(30,44,33,46)(31,45,35,48)

Polycyclic presentation:

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

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, 7622639);