#9243737

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3921 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)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18,23,21)(19,24,29,31)(20,26,30,25)(22,32,28,27)(33,34,40,38)(35,36,45,41)(37,39,47,42)(43,44,48,46);(3,9)(6,13)(10,15)(14,16)(17,19,23,29)(18,24,21,31)(20,22,30,28)(25,27,26,32)(33,35,40,45)(34,41,38,36)(37,43,47,48)(39,46,42,44);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,20,23,30)(18,25,21,26)(19,28,29,22)(24,32,31,27)(33,36)(34,35)(37,44)(38,45)(39,43)(40,41)(42,48)(46,47);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,26)(19,22)(21,25)(23,30)(24,32)(27,31)(28,29)(33,37,40,47)(34,42,38,39)(35,43,45,48)(36,46,41,44);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,21,23,18)(19,24,29,31)(20,25,30,26)(22,32,28,27)(33,38,40,34)(35,41,45,36)(37,39,47,42)(43,44,48,46);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,22)(18,27)(19,20)(21,32)(23,28)(24,25)(26,31)(29,30)(33,39)(34,37)(35,44)(36,43)(38,47)(40,42)(41,48)(45,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=g,e^2=h,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, 9150933);