#8179745

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

4117 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)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34,38,42)(35,40,36,39)(37,41,45,48)(43,47,44,46);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(6,14)(9,12)(11,13)(15,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36,38,35)(34,40,42,39)(37,43,45,44)(41,46,48,47);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

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

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, 8086941);