#8430560

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	#52529	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4142 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)(8,15)(10,16)(13,14)(17,18,23,26)(19,24,21,20)(22,25,31,30)(27,32,29,28)(34,40)(35,44)(36,46)(43,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32)(33,34)(35,41)(36,47)(37,43)(38,44)(39,40)(42,46)(45,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,35)(34,41)(36,45)(37,46)(38,40)(39,44)(42,43)(47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,32)(28,29)(33,36,39,46)(34,42,40,47)(35,45,44,37)(38,48,41,43);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28)(33,37,39,45)(34,43,40,48)(35,46,44,36)(38,47,41,42);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,38)(34,44)(35,40)(36,43)(37,47)(39,41)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,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, 8337756);