#8458076

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	#52422	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4219 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8365272);