#9245130

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	#37471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3860 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=g,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9152326);