#8506299

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	#58197	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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)(3,9)(4,13)(5,16)(6,10)(7,8)(11,15)(12,14)(17,18,22,27)(19,24,28,23)(20,25,31,30)(21,26,32,29)(36,44)(40,45)(43,48)(46,47);(2,8)(5,10)(9,13)(11,14)(19,28)(20,31)(23,24)(25,30)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,46)(47,48);(1,3,7,4)(2,9,8,13)(5,11,10,14)(6,12,16,15)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,32)(26,31)(33,35)(34,37)(36,43)(38,41)(39,42)(40,46)(44,48)(45,47);(1,4,7,3)(2,9,8,13)(5,14,10,11)(6,12,16,15)(17,19)(18,24)(20,29)(21,25)(22,28)(23,27)(26,31)(30,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,48)(45,46);(1,5,7,10)(2,6,8,16)(3,11,4,14)(9,12,13,15)(17,20)(18,25)(19,29)(21,24)(22,31)(23,32)(26,28)(27,30)(33,36,38,44)(34,40,42,45)(35,43,41,48)(37,46,39,47);(1,6)(2,10)(3,12)(4,15)(5,8)(7,16)(9,14)(11,13)(17,21)(18,26)(19,30)(20,24)(22,32)(23,31)(25,28)(27,29)(33,37,38,39)(34,41,42,35)(36,46,44,47)(40,48,45,43)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413495);