#8110597

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

4006 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)(3,10)(4,12)(5,7)(6,8)(9,13)(11,15)(14,16)(17,18,23,27)(19,28,25,21)(20,24,29,32)(22,31,30,26)(33,34,38,41)(35,36,39,44)(37,45,46,40)(42,48,47,43);(2,5)(4,13)(8,10)(11,14)(19,25)(20,29)(21,28)(24,32)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,48)(46,47);(1,3,7,6)(2,8,5,10)(4,11,13,14)(9,15,12,16)(17,19,23,25)(18,21,27,28)(20,26,29,31)(22,24,30,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,47)(42,45)(46,48);(1,4)(2,9)(3,11)(5,12)(6,14)(7,13)(8,15)(10,16)(17,20,23,29)(18,24,27,32)(19,26,25,31)(21,30,28,22);(1,5,7,2)(3,10,6,8)(4,12,13,9)(11,16,14,15)(17,21)(18,25)(19,27)(20,30)(22,29)(23,28)(24,31)(26,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,47)(42,45)(46,48);(1,6,7,3)(2,10,5,8)(4,14,13,11)(9,16,12,15)(17,22,23,30)(18,26,27,31)(19,24,25,32)(20,21,29,28)(33,37,38,46)(34,40,41,45)(35,42,39,47)(36,43,44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017793);