#8434107

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4244 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^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, 8341303);