#8183829

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

4035 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)(3,4)(5,8)(6,14)(7,10)(9,15)(11,13)(12,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34,38,41)(35,37,40,46)(36,39,45,48)(42,44,43,47);(1,3,7,8)(2,4,10,5)(6,11,15,16)(9,12,14,13)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35,38,40)(34,37,41,46)(36,43,45,42)(39,47,48,44);(1,4,7,5)(2,8,10,3)(6,12,15,13)(9,16,14,11)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,35)(34,37)(36,42)(38,40)(39,44)(41,46)(43,45)(47,48);(1,5,7,4)(2,8,10,3)(6,13,15,12)(9,16,14,11)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,43)(41,48)(46,47);(1,6,7,15)(2,9,10,14)(3,11,8,16)(4,12,5,13)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37)(34,40)(35,41)(36,44)(38,46)(39,43)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,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, 8091025);