#8417198

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	#53532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 8324394);