#8436689

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	#54602	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=gh,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, 8343885);