#9814881

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=i,e^2=g,f^2=h,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, 9722077);