#10423047

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	#41717	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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)(3,9)(4,8)(5,7)(6,10)(11,16)(12,15)(13,14)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,34,38,41)(35,42,45,39)(36,37,40,46)(43,48,47,44);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(19,23)(20,27)(26,31)(28,32)(33,34,38,41)(35,39,45,42)(36,37,40,46)(43,44,47,48);(1,3,7,4)(2,9,5,8)(6,11,14,12)(10,16,13,15)(17,19,22,23)(18,20,25,27)(21,26,30,31)(24,28,29,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,32)(28,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,5,7,2)(3,8,4,9)(6,13,14,10)(11,15,12,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,31)(26,29)(30,32)(33,37)(34,40)(35,44)(36,41)(38,46)(39,47)(42,43)(45,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10330243);