#10312366

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	#47089	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,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, 10219562);