#10451611

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3810 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)(3,8)(4,10)(5,7)(6,9)(11,15)(12,16)(13,14)(17,18,23,20)(19,25,21,24)(22,28,30,26)(27,31,29,32)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,48,44,41);(2,5)(8,10)(9,13)(15,16)(17,18,23,20)(19,25,21,24)(22,26,30,28)(27,32,29,31)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,46)(45,47);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,20,23,18)(19,25,21,24)(22,26,30,28)(27,31,29,32)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,6,7,14)(2,9,5,13)(3,11,4,12)(8,15,10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(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=gi,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, 10358807);