#8112413

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	#53679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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)(18,20)(21,29)(23,26)(27,31)(33,34,39,43)(35,36,46,40)(37,41,38,42)(44,47,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(18,20)(19,25)(24,28)(27,31)(33,35,39,46)(34,40,43,36)(37,44,38,45)(41,48,42,47);(2,6)(7,9)(8,13)(14,15)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35)(34,36)(37,44)(38,45)(39,46)(40,43)(41,47)(42,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36)(34,35)(37,47)(38,48)(39,40)(41,44)(42,45)(43,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,37,39,38)(34,41,43,42)(35,44,46,45)(36,47,40,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,38,39,37)(34,42,43,41)(35,45,46,44)(36,48,40,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8019609);