#9938032

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	#37529	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4036 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=gh,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, 9845228);