#8472848

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	#52471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4142 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)(3,9)(4,13)(5,16)(6,10)(7,8)(11,15)(12,14)(20,28)(24,29)(26,32)(30,31)(34,39)(35,41)(40,43)(42,47);(2,8)(5,10)(9,13)(11,14)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(1,3,7,4)(2,9,8,13)(5,11,10,14)(6,12,16,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,4,7,3)(2,9,8,13)(5,14,10,11)(6,12,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,34)(35,37)(36,40)(38,39)(41,46)(42,44)(43,45)(47,48);(1,5,7,10)(2,6,8,16)(3,11,4,14)(9,12,13,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,43)(41,47)(46,48);(1,6)(2,10)(3,12)(4,15)(5,8)(7,16)(9,14)(11,13)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,e^2=gh,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, 8380044);