#8436781

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	#43471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,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, 8343977);