#8893589

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4012 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800785);