#8505965

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	#37332	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 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)(3,7,13,8)(4,9,15,5)(11,12,14,16)(20,28)(24,29)(27,32)(30,31)(33,34)(35,43)(36,42)(37,47)(38,40)(39,41)(44,45)(46,48);(1,3,6,13)(2,7,10,8)(4,14,15,11)(5,12,9,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(33,35,40,39)(34,41,38,43)(36,44,47,46)(37,45,42,48);(1,3)(2,8)(4,11)(5,12)(6,13)(7,10)(9,16)(14,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,36,40,47)(34,42,38,37)(35,44,39,46)(41,48,43,45);(1,3)(2,7)(4,11)(5,16)(6,13)(8,10)(9,12)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,37)(34,36)(35,45)(38,47)(39,48)(40,42)(41,44)(43,46);(1,4)(2,9)(3,11)(5,10)(6,15)(7,12)(8,16)(13,14)(17,20,22,28)(18,24,26,29)(19,27,25,32)(21,30,23,31)(33,38,40,34)(35,41,39,43)(36,37,47,42)(44,48,46,45);(1,5,6,9)(2,4,10,15)(3,12,13,16)(7,14,8,11)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)(33,39,40,35)(34,41,38,43)(36,46,47,44)(37,45,42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,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, 8413161);