#9240954

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	#56948	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3834 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=gh,e^2=gh,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, 9148150);