#8521607

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	#56378	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8428803);