#10451643

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3902 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)(3,5)(4,11)(6,9)(7,8)(10,13)(12,16)(14,15)(17,18,23,25)(19,22,27,32)(20,28,21,24)(26,31,29,30)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,48,44,41);(2,8)(5,11)(9,15)(13,16)(17,19,23,27)(18,22,25,32)(20,29,21,26)(24,31,28,30)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,46)(45,47);(1,3,7,4)(2,5,8,11)(6,10,14,12)(9,13,15,16)(17,20,23,21)(18,24,25,28)(19,26,27,29)(22,30,32,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(1,4,7,3)(2,5,8,11)(6,10,14,12)(9,16,15,13)(17,21,23,20)(18,24,25,28)(19,26,27,29)(22,31,32,30)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,5)(2,4)(3,8)(6,13)(7,11)(9,12)(10,15)(14,16)(17,22)(18,19)(20,30)(21,31)(23,32)(24,26)(25,27)(28,29)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,6,7,14)(2,9,8,15)(3,10,4,12)(5,13,11,16)(17,18)(19,22)(20,24)(21,28)(23,25)(26,30)(27,32)(29,31)(33,34)(35,39)(36,47)(37,41)(38,42)(40,45)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=gi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358839);