#8445710

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352906);