#8418145

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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)(17,18,23,22)(19,27,30,24)(20,25,29,31)(21,32,28,26)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,48)(45,47);(2,3)(4,11)(8,10)(12,16)(17,19,23,30)(18,24,22,27)(20,28,29,21)(25,32,31,26)(33,35)(34,40)(36,38)(37,47)(39,44)(41,45)(42,46)(43,48);(5,13)(8,10)(12,16)(14,15)(18,22)(24,27)(25,31)(26,32)(35,44)(38,45)(40,48)(42,47);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,32)(26,27)(33,36)(34,37)(35,45)(38,44)(39,41)(40,42)(43,46)(47,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,31,27,25)(33,37)(34,41)(35,42)(36,43)(38,40)(39,46)(44,47)(45,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,32,28,26)(33,38,39,45)(34,42,43,47)(35,36,44,41)(37,40,46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,e^2=h,f^2=gi,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, 8325341);