#8093473

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

4151 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)(3,4)(5,13)(6,7)(8,14)(9,11)(10,16)(12,15)(20,28)(24,29)(26,32)(30,31)(33,34)(35,38)(36,39)(37,41)(40,42)(43,46)(44,47)(45,48);(3,9)(4,11)(10,15)(12,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,39,42)(34,40,36,38)(37,45,47,43)(41,46,44,48);(2,7)(3,9)(8,13)(10,15)(18,22)(23,25)(24,29)(30,31)(34,36)(35,42)(41,44)(43,45);(1,3)(2,4)(5,10)(6,9)(7,11)(8,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,39,34)(35,40,42,38)(37,44,47,41)(43,48,45,46);(1,4)(2,3)(5,12)(6,11)(7,9)(8,10)(13,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37,39,47)(34,41,36,44)(35,43,42,45)(38,46,40,48);(1,5,6,14)(2,8,7,13)(3,10,9,15)(4,12,11,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,40)(34,42,36,35)(37,46,47,48)(41,45,44,43)

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

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, 8000669);