#8445765

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	#52594	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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)(3,13)(4,10)(5,7)(6,15)(8,12)(9,16)(11,14)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(34,40)(35,43)(36,45)(42,47);(1,3)(2,9)(4,6)(5,14)(7,15)(8,16)(10,11)(12,13)(19,29)(20,31)(24,28)(25,30)(33,34,39,40)(35,38,43,48)(36,46,45,41)(37,42,44,47);(1,4,8,5)(2,10,12,7)(3,6,16,14)(9,11,13,15)(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,35)(34,38)(36,44)(37,45)(39,43)(40,48)(41,42)(46,47);(1,5,8,4)(2,10,12,7)(3,14,16,6)(9,11,13,15)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,36,39,45)(34,41,40,46)(35,44,43,37)(38,42,48,47);(1,6,8,14)(2,11,12,15)(3,4,16,5)(7,9,10,13)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25)(33,37,39,44)(34,42,40,47)(35,45,43,36)(38,41,48,46);(1,7,8,10)(2,5,12,4)(3,15,16,11)(6,9,14,13)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,38,39,48)(34,43,40,35)(36,47,45,42)(37,41,44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=g,e^2=gi,f^2=g,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, 8352961);