#8107642

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=i,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, 8014838);