#10216815

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=i,e^2=ghi,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, 10124011);