#10320355

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×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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)(3,7)(4,8)(5,9)(6,10)(11,13)(12,15)(14,16)(17,18)(19,21)(20,29)(22,25)(23,28)(24,31)(26,32)(27,30)(33,34,39,37)(35,42,41,38)(36,44,47,40)(43,46,45,48);(1,3,6,8)(2,7,10,4)(5,11,14,15)(9,13,16,12)(17,19,22,28)(18,23,25,21)(20,26,31,27)(24,32,29,30)(33,35,39,41)(34,38,37,42)(36,45,47,43)(40,48,44,46);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,20)(18,24)(19,26)(21,30)(22,31)(23,32)(25,29)(27,28)(33,34,39,37)(35,38,41,42)(36,44,47,40)(43,48,45,46);(1,3)(2,7)(4,10)(5,11)(6,8)(9,13)(12,16)(14,15)(17,20)(18,24)(19,27)(21,32)(22,31)(23,30)(25,29)(26,28)(33,36,39,47)(34,40,37,44)(35,43,41,45)(38,46,42,48);(1,4,6,7)(2,8,10,3)(5,13,14,12)(9,11,16,15)(17,21)(18,19)(20,30)(22,23)(24,26)(25,28)(27,29)(31,32)(33,37,39,34)(35,38,41,42)(36,40,47,44)(43,48,45,46);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,18)(19,23)(20,24)(21,28)(22,25)(26,32)(27,30)(29,31)(33,38)(34,41)(35,37)(36,46)(39,42)(40,45)(43,44)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

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, 10227551);