#8475809

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	#57178	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4221 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8383005);