#8436697

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	#53654	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(34,36)(39,41)(40,44)(46,47);(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,e^2=gh,f^2=g,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, 8343893);