#8621391

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	#47302	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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)(4,10)(5,14)(6,12)(7,9)(11,13)(15,16)(17,18,22,24)(19,26,21,23)(20,28,30,25)(27,31,29,32)(33,34,39,42)(35,37,40,45)(36,38,41,47)(43,44,46,48);(1,3)(2,8)(4,13)(5,15)(6,9)(7,12)(10,11)(14,16)(17,19,22,21)(18,23,24,26)(20,29,30,27)(25,32,28,31)(33,35,39,40)(34,37,42,45)(36,43,41,46)(38,44,47,48);(2,9)(3,12)(5,13)(10,16)(18,24)(23,26)(25,28)(31,32)(35,40)(37,45)(43,46)(44,48);(1,4)(2,10)(3,5)(6,11)(7,15)(8,14)(9,16)(12,13)(17,20,22,30)(18,25,24,28)(19,27,21,29)(23,31,26,32)(33,36)(34,38)(35,43)(37,44)(39,41)(40,46)(42,47)(45,48);(1,5)(2,11)(3,4)(6,10)(7,13)(8,16)(9,14)(12,15)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,30)(29,31)(33,37,39,45)(34,40,42,35)(36,44,41,48)(38,46,47,43);(1,6)(2,12)(3,9)(4,14)(5,10)(7,8)(11,15)(13,16)(17,21,22,19)(18,23,24,26)(20,27,30,29)(25,32,28,31)(33,38)(34,41)(35,44)(36,42)(37,46)(39,47)(40,48)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528587);