#8755347

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	#53841	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(18,23)(21,29)(24,30)(28,32)(33,34)(35,37)(36,44)(38,42)(39,41)(40,46)(43,47)(45,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(20,27)(24,30)(26,31)(28,32)(33,34,38,42)(35,37,41,39)(36,40,46,44)(43,45,48,47);(2,6)(7,9)(8,12)(14,15)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=hi,e^2=gh,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, 8662543);