#8534648

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	#41962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4256 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441844);