#9862973

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	#56616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,23,25)(19,24,22,32)(20,26,29,21)(27,28,30,31)(33,34,38,42)(35,39,37,41)(36,40,45,44)(43,47,46,48);(1,3,7,4)(2,8,9,10)(5,13,15,12)(6,14,16,11)(17,19,23,22)(18,24,25,32)(20,27,29,30)(21,31,26,28)(33,34)(35,39)(36,44)(37,41)(38,42)(40,45)(43,48)(46,47);(2,9)(3,4)(11,14)(12,13)(18,25)(20,29)(24,32)(27,30)(33,35,38,37)(34,39,42,41)(36,43,45,46)(40,47,44,48);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,11,16,14)(17,20)(18,26)(19,27)(21,25)(22,30)(23,29)(24,28)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,5)(2,11)(3,13)(4,12)(6,8)(7,15)(9,14)(10,16)(17,21,23,26)(18,20,25,29)(19,28,22,31)(24,30,32,27)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,48)(42,44);(1,6,7,16)(2,12,9,13)(3,14,4,11)(5,10,15,8)(17,22,23,19)(18,24,25,32)(20,30,29,27)(21,31,26,28)(33,37,38,35)(34,41,42,39)(36,43,45,46)(40,47,44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=i,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9770169);