#8174656

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

4058 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)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(34,36)(37,45)(39,43)(41,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,34,38,36)(35,39,42,43)(37,40,45,44)(41,46,47,48);(2,6)(7,9)(8,13)(14,15)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,36,38,34)(35,39,42,43)(37,44,45,40)(41,46,47,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,37)(34,40)(35,41)(36,44)(38,45)(39,46)(42,47)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081852);