#8436608

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

4084 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343804);