#8445749

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(34,39)(35,42)(36,44)(46,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(19,29)(20,31)(24,28)(25,30)(33,34,38,39)(35,40,42,37)(36,45,44,41)(43,47,48,46);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,35)(34,40)(36,43)(37,39)(38,42)(41,46)(44,48)(45,47);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,36,38,44)(34,41,39,45)(35,43,42,48)(37,47,40,46);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,37,38,40)(34,35,39,42)(36,46,44,47)(41,48,45,43)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=g,e^2=i,f^2=g,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, 8352945);