#9240939

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3797 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=gh,e^2=h,f^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, 9148135);