#8734360

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8641556);