#8461985

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	#35044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4108 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=ghi,f^2=ghi,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, 8369181);