#7715515

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4443 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)(3,7)(4,5)(6,8)(10,13)(14,16)(17,18,23,19)(20,24,30,27)(21,25,34,28)(22,29,35,26)(31,36,42,39)(32,40,43,37)(33,41,45,38)(44,47,48,46);(9,10)(11,14)(12,13)(15,16)(18,19)(20,30)(25,28)(26,29)(31,42)(32,43)(38,41)(44,48);(2,5)(6,7)(9,11,12,15)(10,14,13,16)(17,19,23,18)(20,27,30,24)(21,25,34,28)(22,29,35,26)(31,36,42,39)(32,40,43,37)(33,38,45,41)(44,46,48,47);(1,2)(3,6)(4,5)(7,8)(17,20)(18,24)(19,27)(21,31)(22,32)(23,30)(25,36)(26,37)(28,39)(29,40)(33,44)(34,42)(35,43)(38,46)(41,47)(45,48);(1,3,4,8)(2,6,5,7)(17,21)(18,25)(19,28)(20,31)(22,33)(23,34)(24,36)(26,38)(27,39)(29,41)(30,42)(32,44)(35,45)(37,46)(40,47)(43,48);(17,22,23,35)(18,26,19,29)(20,32,30,43)(21,33,34,45)(24,37,27,40)(25,38,28,41)(31,44,42,48)(36,46,39,47)

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=h,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, 7622711);