#8932828

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	#35473	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=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, 8840024);