#8690992

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 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)(3,6)(4,10)(5,14)(7,11)(8,12)(9,13)(15,16)(18,20)(19,25)(24,28)(27,29)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(1,3,8,13)(2,9,12,6)(4,7,5,15)(10,16,14,11)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(35,44)(36,45)(40,46)(41,48);(1,4,8,5)(2,10,12,14)(3,7,13,15)(6,11,9,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,35)(34,40)(36,37)(38,41)(39,44)(42,45)(43,46)(47,48);(1,5,8,4)(2,10,12,14)(3,7,13,15)(6,16,9,11)(17,20,22,18)(19,26,25,23)(21,24,30,28)(27,31,29,32)(33,36)(34,41)(35,37)(38,40)(39,45)(42,44)(43,48)(46,47);(1,6)(2,3)(4,11)(5,16)(7,10)(8,9)(12,13)(14,15)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,37)(34,38)(35,36)(39,42)(40,41)(43,47)(44,45)(46,48);(1,7)(2,11)(3,5)(4,13)(6,14)(8,15)(9,10)(12,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38)(34,42)(35,41)(36,46)(37,43)(39,47)(40,45)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8598188);