#10325129

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	#57203	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10232325);