#8754997

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

3771 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:

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

Polycyclic presentation:

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

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, 8662193);