#8436685

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	#41138	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(18,24)(19,20)(26,31)(27,28)(33,34)(35,40)(36,47)(37,41)(38,42)(39,43)(44,45)(46,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,18)(19,25)(20,22)(21,26)(23,24)(27,32)(28,29)(30,31)(33,35,39,46)(34,40,43,48)(36,37,42,44)(38,45,47,41);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,19,23,20)(18,25,24,22)(21,27,30,28)(26,32,31,29)(33,36,39,42)(34,38,43,47)(35,44,46,37)(40,41,48,45);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,20,23,19)(18,25,24,22)(21,28,30,27)(26,32,31,29)(33,35)(34,40)(36,44)(37,42)(38,41)(39,46)(43,48)(45,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,21,23,30)(18,26,24,31)(19,27,20,28)(22,29,25,32)(33,37,39,44)(34,41,43,45)(35,42,46,36)(38,48,47,40);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,22,23,25)(18,20,24,19)(21,29,30,32)(26,28,31,27)(33,38)(34,42)(35,45)(36,43)(37,48)(39,47)(40,44)(41,46)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,e^2=h,f^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, 8343881);