#8734382

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	#57178	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4232 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8641578);