#8505989

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=hi,f^2=gi,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, 8413185);