#8472386

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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)(19,27)(20,26)(21,31)(22,24)(23,25)(28,29)(30,32)(33,34)(35,43)(36,47)(37,48)(38,42)(39,40)(41,44)(45,46);(1,2)(3,7)(4,13)(5,9)(6,11)(8,12)(10,14)(15,16)(17,19,24,23)(18,25,22,27)(20,28,31,30)(21,29,26,32)(34,40)(35,44)(36,48)(42,46);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,16)(13,15)(17,20,24,31)(18,26,22,21)(19,28,23,30)(25,32,27,29)(33,35,39,44)(34,41,40,43)(36,45,48,38)(37,46,47,42);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,21)(18,20)(19,29)(22,31)(23,32)(24,26)(25,28)(27,30)(33,36)(34,37)(35,45)(38,44)(39,48)(40,47)(41,46)(42,43);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,13)(11,16)(17,22,24,18)(19,25,23,27)(20,21,31,26)(28,32,30,29)(33,37)(34,36)(35,46)(38,43)(39,47)(40,48)(41,45)(42,44);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,23,24,19)(18,25,22,27)(20,30,31,28)(21,29,26,32)(33,38,39,45)(34,42,40,46)(35,36,44,48)(37,43,47,41)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=gh,f^2=g,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, 8379582);