#9938228

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	#37527	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,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, 9845424);