#9938191

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	#56264	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

(1,2,6,7)(3,8,11,15)(4,9,14,13)(5,12,16,10)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(33,34)(35,40)(36,46)(37,42)(38,47)(39,41)(43,45)(44,48);(2,7)(3,11)(9,13)(10,12)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35,39,44)(34,40,41,48)(36,38,42,45)(37,43,46,47);(2,7)(3,11)(9,13)(10,12)(18,20)(23,25)(24,28)(30,31)(34,41)(35,44)(36,42)(43,47);(1,3)(2,8)(4,12)(5,13)(6,11)(7,15)(9,16)(10,14)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,36,39,42)(34,37,41,46)(35,45,44,38)(40,47,48,43);(1,4)(2,9)(3,12)(5,15)(6,14)(7,13)(8,16)(10,11)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,37)(34,42)(35,43)(36,41)(38,48)(39,46)(40,45)(44,47);(1,5,6,16)(2,10,7,12)(3,13,11,9)(4,8,14,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(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=gh,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, 9845387);