#10195408

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3844 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 10102604);