#8690991

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

4145 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 8598187);