#8137129

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	#57210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4124 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)(3,12)(4,5)(6,14)(7,11)(8,16)(9,15)(10,13)(20,28)(24,31)(26,32)(29,30)(33,34)(35,41)(36,45)(37,46)(38,40)(39,44)(42,43)(47,48);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,34,38,40)(35,39,44,41)(36,45,46,37)(42,47,48,43);(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,41)(45,47)(46,48);(1,4,7,15)(2,9,11,5)(3,13,16,6)(8,10,12,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,46)(34,37,40,45)(35,42,44,48)(39,43,41,47);(1,5,7,9)(2,4,11,15)(3,14,16,10)(6,8,13,12)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,37)(34,36)(35,43)(38,45)(39,42)(40,46)(41,48)(44,47);(1,6,7,13)(2,10,11,14)(3,15,16,4)(5,8,9,12)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,41)(45,47)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8044325);