#8926533

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3892 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,e^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, 8833729);