#8107213

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	#52476	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,16)(12,15)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(34,39)(35,44)(36,46)(37,47);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(18,19)(23,25)(24,26)(30,31)(33,34)(35,40)(36,42)(37,43)(38,39)(41,44)(45,46)(47,48);(2,7)(3,8)(9,13)(11,15)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35,38,44)(34,40,39,41)(36,43,46,48)(37,45,47,42);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,41)(36,43)(37,45)(38,44)(39,40)(42,47)(46,48);(1,4,6,10)(2,8,7,3)(5,12,14,16)(9,15,13,11)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36)(34,42)(35,43)(37,40)(38,46)(39,45)(41,47)(44,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,47)(34,43,39,48)(35,45,44,42)(36,40,46,41)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014409);