#8445974

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

3985 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:

(1,2,7,10)(3,11,4,8)(5,6,9,16)(12,15,14,13)(17,18,23,25)(19,26,30,24)(20,22,21,31)(27,32,28,29)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(3,4)(8,11)(12,14)(13,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,3,7,4)(2,8,10,11)(5,12,9,14)(6,13,16,15)(17,20,23,21)(18,22,25,31)(19,27,30,28)(24,29,26,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,4,7,3)(2,8,10,11)(5,14,9,12)(6,13,16,15)(17,21,23,20)(18,22,25,31)(19,28,30,27)(24,29,26,32)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(1,5,7,9)(2,6,10,16)(3,12,4,14)(8,13,11,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32);(1,6)(2,9)(3,13)(4,15)(5,10)(7,16)(8,14)(11,12)(17,22)(18,21)(19,29)(20,25)(23,31)(24,28)(26,27)(30,32)(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,d^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, 8353170);