#9938145

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	#55695	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9845341);