#8131224

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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:

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

Polycyclic presentation:

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

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, 8038420);