#10188289

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	#50024	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(17,18,22,25)(19,23,29,32)(20,24,30,21)(26,28,27,31)(34,39)(35,41)(37,45)(44,46);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,16)(10,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,32)(26,30)(33,34,38,39)(35,40,41,36)(37,42,45,47)(43,46,48,44);(17,20)(18,24)(19,26)(21,25)(22,30)(23,28)(27,29)(31,32)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,46,47,44);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,16,11,10)(17,20)(18,24)(19,27)(21,25)(22,30)(23,31)(26,29)(28,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,46)(44,47)(45,48);(1,5,7,2)(3,11,14,8)(4,9,13,15)(6,10,12,16)(17,19)(18,23)(20,26)(21,31)(22,29)(24,28)(25,32)(27,30)(33,36,38,40)(34,41,39,35)(37,46,45,44)(42,43,47,48);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,15,11,9)(17,21,22,24)(18,20,25,30)(19,28,29,31)(23,27,32,26)(33,37)(34,42)(35,43)(36,44)(38,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=ghi,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, 10095485);