#8095166

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	#45610	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4111 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)(3,8)(4,10)(5,13)(6,7)(9,14)(11,12)(15,16)(18,21)(19,25)(24,29)(26,31)(33,34)(35,41)(36,45)(37,42)(38,39)(40,43)(44,46)(47,48);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,15)(14,16)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,35,40,38)(34,41,43,39)(36,44,37,47)(42,48,45,46);(2,7)(3,10)(9,13)(11,16)(20,28)(24,29)(26,31)(30,32)(33,36,40,37)(34,42,43,45)(35,44,38,47)(39,46,41,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,37,40,36)(34,42,43,45)(35,44,38,47)(39,48,41,46);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,38,40,35)(34,39,43,41)(36,44,37,47)(42,48,45,46);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,21,22,18)(19,27,25,23)(20,29,28,24)(26,32,31,30)(33,39)(34,35)(36,46)(37,48)(38,43)(40,41)(42,44)(45,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002362);