#8179648

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

4012 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gh,e^2=h,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, 8086844);