#8141448

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	#37891	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4214 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)(3,10)(4,13)(5,8)(6,14)(7,9)(11,12)(15,16)(17,18)(19,26)(20,24)(21,23)(22,29)(25,30)(27,31)(28,32)(37,45)(40,47)(43,44)(46,48);(1,3,7,5)(2,8,9,10)(4,11,14,15)(6,12,13,16)(19,20)(24,26)(27,28)(31,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,47,48);(1,4)(2,6)(3,11)(5,15)(7,14)(8,12)(9,13)(10,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,47,48);(1,5,7,3)(2,8,9,10)(4,11,14,15)(6,16,13,12)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31)(33,34)(35,39)(36,42)(37,40)(38,41)(43,46)(44,48)(45,47);(1,6)(2,4)(3,12)(5,16)(7,13)(8,11)(9,14)(10,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8048644);