#8504766

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	#59779	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=i,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8411962);