#8459429

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	#37249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4174 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)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(20,28)(24,31)(27,32)(29,30)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47);(1,3)(2,8)(4,6)(5,13)(7,14)(9,10)(11,12)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(35,44)(36,46)(39,43)(40,47);(1,4,7,5)(2,9,11,15)(3,6,14,13)(8,10,12,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,41,47,48);(1,5,7,4)(2,9,11,15)(3,13,14,6)(8,10,12,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,40)(35,45)(37,44)(38,46)(39,41)(42,47)(43,48);(1,6,7,13)(2,10,11,16)(3,4,14,5)(8,9,12,15)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,38,45)(34,41,42,48)(35,36,44,46)(39,47,43,40);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,34)(35,39)(36,47)(37,41)(38,42)(40,46)(43,44)(45,48)

Polycyclic presentation:

2^9|c^2=gh,d^2=g,e^2=gi,f^2=hi,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, 8366625);