#10195407

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3767 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10102603);