#10325147

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	#56438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 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)(18,24)(19,20)(22,26)(29,31)(33,34)(35,38)(36,41)(37,42)(39,43)(40,45)(44,46)(47,48);(3,11)(5,15)(7,16)(9,12)(17,18)(19,28)(20,25)(21,26)(22,27)(23,24)(29,30)(31,32)(33,35,39,45)(34,40,43,38)(36,44,47,37)(41,42,48,46);(1,3)(2,7)(4,9)(5,13)(6,11)(8,15)(10,16)(12,14)(17,19,23,20)(18,25,24,28)(21,29,27,31)(22,30,26,32)(33,36,39,47)(34,41,43,48)(35,44,45,37)(38,46,40,42);(1,3)(2,7)(4,9)(5,13)(6,11)(8,15)(10,16)(12,14)(17,20,23,19)(18,25,24,28)(21,29,27,31)(22,32,26,30)(33,35)(34,40)(36,44)(37,47)(38,43)(39,45)(41,42)(46,48);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,21,23,27)(18,26,24,22)(19,29,20,31)(25,32,28,30)(33,37,39,44)(34,42,43,46)(35,36,45,47)(38,41,40,48);(1,5)(2,9)(3,13)(4,7)(6,15)(8,11)(10,12)(14,16)(17,22,23,26)(18,27,24,21)(19,30,20,32)(25,31,28,29)(33,38,39,40)(34,35,43,45)(36,46,47,42)(37,41,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=g,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, 10232343);