#10521308

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	#45727	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4273 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428504);