#10521284

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

3805 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)(3,10,6,8)(5,9,13,12)(11,16,14,15)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34)(35,41)(36,42)(37,48)(38,43)(39,44)(40,45)(46,47);(1,3,7,6)(2,8,4,10)(5,11,13,14)(9,15,12,16)(33,35,40,38)(34,41,45,43)(36,44,46,37)(39,47,48,42);(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36)(34,42)(35,37)(38,44)(39,43)(40,46)(41,48)(45,47);(1,4,7,2)(3,8,6,10)(5,12,13,9)(11,15,14,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,37,40,44)(34,39,45,48)(35,46,38,36)(41,42,43,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,38,40,35)(34,43,45,41)(36,44,46,37)(39,47,48,42);(1,6,7,3)(2,8,4,10)(5,11,13,14)(9,16,12,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,39)(34,44)(35,47)(36,43)(37,45)(38,42)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428480);