#8322240

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3927 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8229436);