#9911752

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	#55328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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)(3,11)(4,10)(5,8)(6,14)(7,9)(12,13)(15,16)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34,39,37)(35,40,36,41)(38,42,46,45)(43,47,44,48);(1,3,7,5)(2,8,9,11)(4,12,14,15)(6,13,10,16)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(34,37)(40,41)(42,45)(47,48);(2,9)(4,14)(8,11)(12,15)(18,25)(19,29)(20,31)(27,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,4)(2,10)(3,12)(5,15)(6,9)(7,14)(8,16)(11,13)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,5,7,3)(2,8,9,11)(4,15,14,12)(6,13,10,16)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,37,39,34)(35,41,36,40)(38,42,46,45)(43,47,44,48);(1,6,7,10)(2,4,9,14)(3,13,5,16)(8,12,11,15)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219281, #72219283
maximal quotient of	#72219272

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9818948);