#8926719

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(5,13)(6,9)(8,14)(11,15)(12,16)(18,22)(23,28)(24,29)(30,32)(33,34,38,39)(35,42,37,40)(36,41,44,46)(43,48,45,47);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(20,27)(24,29)(25,31)(30,32)(34,39)(36,44)(40,42)(43,45);(1,3,6,4)(2,7,9,10)(5,12,14,11)(8,16,13,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35,38,37)(34,40,39,42)(36,43,44,45)(41,47,46,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,37,38,35)(34,42,39,40)(36,45,44,43)(41,48,46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833915);