#8108779

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	#53191	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(5,13)(6,7)(8,11)(9,14)(10,15)(12,16)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(34,39)(35,43)(36,46)(37,47);(2,7)(4,8)(9,13)(12,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,34,38,39)(35,40,43,48)(36,45,46,41)(37,42,47,44);(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,35)(34,40)(36,44)(37,45)(38,43)(39,48)(41,47)(42,46);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,13)(14,16)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,36,38,46)(34,41,39,45)(35,44,43,42)(37,40,47,48);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,47)(45,46);(1,5,6,14)(2,9,7,13)(3,10,11,16)(4,12,8,15)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,37,38,47)(34,42,39,44)(35,45,43,41)(36,40,46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015975);