#8112545

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(21,29)(25,30)(26,31)(27,32)(34,38)(36,43)(39,44)(42,47);(2,3)(4,11)(8,10)(12,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(35,41)(39,44)(42,47)(46,48);(2,3)(7,9)(8,10)(14,15)(18,23)(19,24)(25,30)(26,31)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)

Polycyclic presentation:

2^9|c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019741);