#8108775

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

3994 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=h,f^2=g,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, 8015971);