#9243505

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3715 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=h,e^2=g,f^2=g,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, 9150701);