#8479554

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386750);