#10521324

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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:

(1,2,6,9)(3,7,11,4)(5,8,14,16)(10,13,15,12)(17,18,24,22)(19,21,25,20)(23,30,31,26)(27,28,32,29)(33,34)(35,41)(36,42)(37,48)(38,43)(39,44)(40,45)(46,47);(1,3,6,11)(2,4,9,7)(5,10,14,15)(8,12,16,13)(17,19)(18,20)(21,22)(23,27)(24,25)(26,28)(29,30)(31,32)(33,35,40,38)(34,41,45,43)(36,44,46,37)(39,47,48,42);(17,20,24,21)(18,25,22,19)(23,28,31,29)(26,32,30,27)(33,36)(34,42)(35,37)(38,44)(39,43)(40,46)(41,48)(45,47);(1,3)(2,7)(4,9)(5,10)(6,11)(8,13)(12,16)(14,15)(17,21,24,20)(18,25,22,19)(23,29,31,28)(26,32,30,27)(33,37,40,44)(34,39,45,48)(35,46,38,36)(41,42,43,47);(1,4,6,7)(2,3,9,11)(5,13,14,12)(8,15,16,10)(17,22,24,18)(19,21,25,20)(23,26,31,30)(27,28,32,29)(33,38,40,35)(34,43,45,41)(36,44,46,37)(39,47,48,42);(1,5)(2,8)(3,10)(4,12)(6,14)(7,13)(9,16)(11,15)(17,23)(18,26)(19,27)(20,28)(21,29)(22,30)(24,31)(25,32)(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=ghi,c^2=h,d^2=hi,e^2=gi,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, 10428520);