#9815123

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3818 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18,23,25)(19,24,28,21)(20,26,30,32)(22,31,29,27)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(2,7)(4,8)(9,15)(12,13)(17,19,23,28)(18,24,25,21)(20,29,30,22)(26,27,32,31)(35,41)(39,42)(43,45)(47,48);(18,25)(19,28)(20,30)(27,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,3)(2,8)(4,7)(5,10)(6,11)(9,13)(12,15)(14,16)(17,20)(18,26)(19,29)(21,31)(22,28)(23,30)(24,27)(25,32)(33,36,38,34)(35,42,41,39)(37,44,46,40)(43,48,45,47);(1,4,6,8)(2,3,7,11)(5,13,14,12)(9,16,15,10)(17,21)(18,19)(20,31)(22,32)(23,24)(25,28)(26,29)(27,30)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,22,23,29)(18,27,25,31)(19,20,28,30)(21,26,24,32)(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=h,d^2=hi,e^2=g,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, 9722319);