#8460668

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	#53416	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=g,e^2=gi,f^2=hi,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, 8367864);