#10451613

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=gi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10358809);