#8115433

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	#42591	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3859 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)(3,12)(4,5)(6,8)(7,9)(10,11)(13,16)(14,15)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(33,34,39,37)(35,40,36,43)(38,45,46,41)(42,48,44,47);(1,3,7,6)(2,8,9,12)(4,15,11,13)(5,16,10,14)(18,23)(19,24)(25,30)(26,31)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(2,9)(4,11)(8,12)(13,15)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,4,7,11)(2,10,9,5)(3,13,6,15)(8,16,12,14)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,5,7,10)(2,11,9,4)(3,14,6,16)(8,15,12,13)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48);(1,6,7,3)(2,12,9,8)(4,15,11,13)(5,16,10,14)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gi,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, 8022629);