#9039157

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	#49622	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

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, 8946353);