#8417218

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	#54161	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=gi,e^2=i,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, 8324414);