#8417232

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	#52646	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324428);