#8211697

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	#44570	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

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

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, 8118893);