#8434331

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341527);