#9910962

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	#53937	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=h,f^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9818158);