#8734373

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	#52316	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)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,34,39,41)(35,44,38,40)(36,42,46,48)(37,45,47,43);(1,3)(2,5)(4,7)(6,10)(8,11)(9,13)(12,15)(14,16)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,32)(29,31)(33,35,39,38)(34,40,41,44)(36,43,46,45)(37,42,47,48);(2,8)(3,4)(9,16)(10,12)(20,28)(23,31)(27,32)(29,30)(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)(18,23)(19,27)(21,30)(22,28)(24,29)(25,32)(26,31)(33,36,39,46)(34,42,41,48)(35,45,38,43)(37,44,47,40);(1,5)(2,3)(4,8)(6,14)(7,11)(9,12)(10,16)(13,15)(17,19)(18,24)(20,27)(21,26)(22,25)(23,29)(28,32)(30,31)(33,37)(34,43)(35,42)(36,44)(38,48)(39,47)(40,46)(41,45);(1,6)(2,9)(3,10)(4,12)(5,13)(7,15)(8,16)(11,14)(17,21,22,24)(18,25,26,19)(20,29,28,30)(23,27,31,32)(33,38,39,35)(34,40,41,44)(36,45,46,43)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,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, 8641569);