#10482480

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,24,27)(19,28,22,25)(20,23,21,26)(29,30,31,32)(33,34,38,41)(35,42,37,39)(36,40,45,48)(43,47,44,46);(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,19,24,22)(18,25,27,28)(20,31,21,29)(23,30,26,32)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(17,20,24,21)(18,26,27,23)(19,29,22,31)(25,30,28,32)(33,35,38,37)(34,39,41,42)(36,43,45,44)(40,46,48,47);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,21,24,20)(18,23,27,26)(19,29,22,31)(25,30,28,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,48)(42,47);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,22,24,19)(18,25,27,28)(20,31,21,29)(23,32,26,30)(33,37,38,35)(34,39,41,42)(36,43,45,44)(40,47,48,46);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,23)(18,20)(19,30)(21,27)(22,32)(24,26)(25,31)(28,29)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10389676);