#8445655

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	#35056	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(34,39)(35,41)(36,44)(43,48);(1,3,7,14)(2,8,11,12)(4,13,5,6)(9,16,15,10)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,34,38,39)(35,40,41,46)(36,45,44,37)(42,48,47,43);(1,4,7,5)(2,9,11,15)(3,13,14,6)(8,16,12,10)(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,35)(34,40)(36,42)(37,43)(38,41)(39,46)(44,47)(45,48);(1,5,7,4)(2,9,11,15)(3,6,14,13)(8,16,12,10)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,36,38,44)(34,37,39,45)(35,42,41,47)(40,43,46,48);(1,6)(2,10)(3,4)(5,14)(7,13)(8,9)(11,16)(12,15)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,37)(34,36)(35,43)(38,45)(39,44)(40,42)(41,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^2=i,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, 8352851);