#9714768

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	#54921	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,20)(21,29)(22,23)(24,26)(25,31)(27,30)(28,32)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(3,9)(7,13)(10,15)(14,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,30)(31,32)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48);(2,6)(7,13)(8,11)(14,16)(18,23)(19,26)(25,29)(27,32)(33,36,38,42)(34,39,41,35)(37,44,46,47)(40,45,48,43);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,30)(28,29)(31,32)(33,36)(34,35)(37,44)(38,42)(39,41)(40,43)(45,48)(46,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,32,29,27)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,48)(42,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,31)(18,25,23,29)(19,27,26,32)(20,28,24,30)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9621964);