#10325396

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

4122 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,e^2=g,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, 10232592);