#8174369

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,23,27)(19,24,29,32)(20,21,25,30)(22,31,28,26)(33,34)(35,41)(36,45)(37,40)(38,47)(39,42)(43,44)(46,48);(2,3)(7,9)(8,10)(14,15)(17,19,23,29)(18,24,27,32)(20,28,25,22)(21,26,30,31)(33,35,39,37)(34,40,42,41)(36,43,47,46)(38,44,45,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20,23,25)(18,21,27,30)(19,22,29,28)(24,31,32,26)(33,35,39,37)(34,41,42,40)(36,46,47,43)(38,44,45,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,21)(19,28)(22,29)(23,25)(24,26)(27,30)(31,32)(33,36)(34,38)(35,43)(37,46)(39,47)(40,44)(41,48)(42,45);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21)(18,25)(19,26)(20,27)(22,24)(23,30)(28,32)(29,31)(33,37,39,35)(34,40,42,41)(36,43,47,46)(38,48,45,44);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,28)(18,26,27,31)(19,20,29,25)(21,24,30,32)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,46)(42,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081565);