#8202208

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8109404);