#8107290

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,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,d^c=dgi

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, 8014486);