#10325810

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3800 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10233006);