#8097322

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	#52317	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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)(4,11)(5,13)(7,9)(8,16)(10,12)(14,15)(17,18,23,28)(19,22,27,24)(20,25,30,32)(21,31,29,26)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(1,3)(2,4)(5,14)(6,9)(7,11)(8,15)(10,16)(12,13)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,31)(26,32)(33,34)(35,39)(36,47)(37,41)(38,42)(40,45)(43,46)(44,48);(1,4,7,6)(2,3,9,11)(5,12,16,15)(8,10,13,14)(17,20,23,30)(18,25,28,32)(19,29,27,21)(22,26,24,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(1,5)(2,8)(3,10)(4,12)(6,15)(7,16)(9,13)(11,14)(17,21,23,29)(18,26,28,31)(19,30,27,20)(22,25,24,32)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,6,7,4)(2,3,9,11)(5,15,16,12)(8,10,13,14)(17,19)(18,22)(20,29)(21,30)(23,27)(24,28)(25,26)(31,32)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,3)(2,6)(4,9)(5,10)(7,11)(8,15)(12,13)(14,16)(17,22)(18,27)(19,28)(20,26)(21,25)(23,24)(29,32)(30,31)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,46)(45,47)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004518);