#10188288

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

3846 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,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, 10095484);