#8752446

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	#48933	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=gh,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, 8659642);