#8110458

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gi,e^2=ghi,f^2=h,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, 8017654);