#8757304

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	#56864	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664500);