#10195731

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	#49245	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=g,e^2=gh,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, 10102927);