#10328271

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,15)(12,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(34,39)(36,43)(40,45)(44,47);(2,7)(3,10)(5,14)(12,15)(19,27)(21,31)(23,32)(25,28)(33,34,38,39)(35,36,42,43)(37,45,46,40)(41,47,48,44);(2,7)(4,8)(9,13)(12,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,35,38,42)(34,36,39,43)(37,41,46,48)(40,44,45,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36,38,43)(34,35,39,42)(37,44,46,47)(40,41,45,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,35)(34,36)(37,41)(38,42)(39,43)(40,44)(45,47)(46,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,37)(34,40)(35,41)(36,44)(38,46)(39,45)(42,48)(43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=h,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, 10235467);