#8093358

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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:

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

Polycyclic presentation:

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

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, 8000554);