#10128022

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	#44152	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3992 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)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34,39,42)(35,40,37,38)(36,41,46,48)(43,47,44,45);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,16)(10,15)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35,39,37)(34,40,42,38)(36,44,46,43)(41,45,48,47);(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,36)(34,41)(35,43)(37,44)(38,45)(39,46)(40,47)(42,48);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,16,11,10)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,37,39,35)(34,38,42,40)(36,44,46,43)(41,45,48,47);(1,5,7,2)(3,11,14,8)(4,9,13,15)(6,10,12,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,36)(34,41)(35,44)(37,43)(38,47)(39,46)(40,45)(42,48);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,15,11,9)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,38)(34,35)(36,45)(37,42)(39,40)(41,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=gi,e^2=g,f^2=g,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, 10035218);