#8898854

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

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, 8806050);