#8472842

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=gh,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, 8380038);