#8757463

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	#56269	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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)(3,8)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(34,36)(35,43)(38,48)(41,45);(1,3,7,4)(2,8,9,5)(6,12,15,11)(10,13,14,16)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(33,34,39,36)(35,44,43,40)(37,45,47,41)(38,42,48,46);(2,9)(3,4)(10,14)(11,12)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35)(34,40)(36,44)(37,38)(39,43)(41,42)(45,46)(47,48);(1,4,7,3)(2,5,9,8)(6,12,15,11)(10,13,14,16)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,36,39,34)(35,44,43,40)(37,45,47,41)(38,46,48,42);(1,5)(2,3)(4,9)(6,13)(7,8)(10,11)(12,14)(15,16)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,37)(34,41)(35,38)(36,45)(39,47)(40,42)(43,48)(44,46);(1,6,7,15)(2,10,9,14)(3,11,4,12)(5,13,8,16)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38)(34,42)(35,37)(36,46)(39,48)(40,41)(43,47)(44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=h,e^2=hi,f^2=gh,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, 8664659);