#8436612

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3992 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=h,e^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, 8343808);