#10038552

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	#44234	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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)(3,6)(4,12)(5,7)(8,10)(9,13)(11,15)(14,16)(19,25)(23,26)(27,31)(30,32)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,48,44,42);(1,3,5,10)(2,6,7,8)(4,11,13,16)(9,14,12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,42,46,48);(2,7)(3,10)(9,12)(11,16)(18,22)(23,26)(24,28)(30,32)(33,36,39,47)(34,41,37,46)(35,38,45,44)(40,48,43,42);(1,3)(2,8)(4,11)(5,10)(6,7)(9,15)(12,14)(13,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,42)(43,48);(1,2)(3,8)(4,12)(5,7)(6,10)(9,13)(11,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,37,39,34)(35,40,45,43)(36,46,47,41)(38,48,44,42);(1,4,5,13)(2,9,7,12)(3,11,10,16)(6,14,8,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,46,43,41)

Polycyclic presentation:

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

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, 9945748);