#10171904

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3848 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10079100);