#8108036

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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)(7,14)(10,16)(13,15)(18,22)(20,28)(23,25)(26,32)(33,34)(35,43)(36,46)(37,42)(38,41)(39,40)(44,47)(45,48);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(34,40)(36,43)(41,42)(47,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,39,46)(34,36,40,43)(37,44,38,45)(41,47,42,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36)(34,35)(37,47)(38,48)(39,43)(40,46)(41,44)(42,45);(33,37,39,38)(34,41,40,42)(35,44,46,45)(36,47,43,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38,39,37)(34,42,40,41)(35,45,46,44)(36,48,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015232);