#8285496

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

3956 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8192692);