#10521336

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

3780 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:

(1,2,6,3)(4,7,12,9)(5,8,14,10)(11,15,13,16)(17,18)(19,24)(20,30)(21,25)(22,31)(23,27)(26,28)(29,32)(33,34)(35,41)(36,42)(37,48)(38,43)(39,44)(40,45)(46,47);(2,3)(7,9)(8,10)(15,16)(17,19,23,29)(18,24,27,32)(20,28,21,22)(25,31,30,26)(33,35,40,38)(34,41,45,43)(36,44,46,37)(39,47,48,42);(17,20,23,21)(18,25,27,30)(19,28,29,22)(24,31,32,26)(33,36)(34,42)(35,37)(38,44)(39,43)(40,46)(41,48)(45,47);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,21,23,20)(18,25,27,30)(19,22,29,28)(24,31,32,26)(33,37,40,44)(34,39,45,48)(35,46,38,36)(41,42,43,47);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,31)(29,32)(33,38,40,35)(34,43,45,41)(36,44,46,37)(39,47,48,42);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22)(18,26)(19,20)(21,29)(23,28)(24,25)(27,31)(30,32)(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=hi,c^2=h,d^2=ghi,e^2=gi,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, 10428532);