#10521333

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

3855 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)(3,8)(4,14)(5,9)(6,15)(7,11)(10,12)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,34)(35,41)(36,42)(37,48)(38,43)(39,44)(40,45)(46,47);(1,3,7,13)(2,8,11,16)(4,12,5,6)(9,15,14,10)(18,23)(24,25)(26,30)(31,32)(33,35,40,38)(34,41,45,43)(36,44,46,37)(39,47,48,42);(1,4,7,5)(2,9,11,14)(3,12,13,6)(8,15,16,10)(17,19,22,20)(18,24,23,25)(21,27,29,28)(26,31,30,32)(33,36)(34,42)(35,37)(38,44)(39,43)(40,46)(41,48)(45,47);(1,5,7,4)(2,9,11,14)(3,6,13,12)(8,15,16,10)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,37,40,44)(34,39,45,48)(35,46,38,36)(41,42,43,47);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,20,22,19)(18,25,23,24)(21,27,29,28)(26,31,30,32)(33,38,40,35)(34,43,45,41)(36,44,46,37)(39,47,48,42);(1,6)(2,10)(3,4)(5,13)(7,12)(8,9)(11,15)(14,16)(17,21,22,29)(18,26,23,30)(19,27,20,28)(24,31,25,32)(33,39)(34,44)(35,47)(36,43)(37,45)(38,42)(40,48)(41,46)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=hi,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, 10428529);