#8893604

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=gh,e^2=g,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, 8800800);