#8436723

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	#41639	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(17,18,22,24)(19,25,27,21)(20,23,29,30)(26,31,32,28)(33,34,39,43)(35,38,42,47)(36,46,45,40)(37,41,44,48);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,19,22,27)(18,21,24,25)(20,26,29,32)(23,28,30,31)(33,35,39,42)(34,38,43,47)(36,37,45,44)(40,48,46,41);(17,18)(19,21)(20,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,36,39,45)(34,40,43,46)(35,44,42,37)(38,41,47,48);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,18)(19,25)(20,23)(21,27)(22,24)(26,31)(28,32)(29,30)(33,35)(34,38)(36,44)(37,45)(39,42)(40,41)(43,47)(46,48);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,20,22,29)(18,23,24,30)(19,26,27,32)(21,28,25,31)(33,37,39,44)(34,41,43,48)(35,45,42,36)(38,40,47,46);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,21,22,25)(18,19,24,27)(20,28,29,31)(23,26,30,32)(33,38)(34,42)(35,43)(36,41)(37,40)(39,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,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, 8343919);