#8472382

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	#39110	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(3,10)(4,12)(5,7)(6,8)(9,13)(11,15)(14,16)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30)(36,44)(40,45)(42,48)(46,47);(1,3,7,6)(2,8,5,10)(4,11,13,14)(9,15,12,16)(18,23)(19,26)(24,30)(27,32)(33,34)(35,39)(36,45)(37,41)(38,43)(40,44)(42,46)(47,48);(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,4)(2,9)(3,11)(5,12)(6,14)(7,13)(8,15)(10,16)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,5,7,2)(3,8,6,10)(4,12,13,9)(11,15,14,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,45)(43,48);(1,6,7,3)(2,8,5,10)(4,11,13,14)(9,16,12,15)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|b^2=gh,e^2=gh,f^2=g,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, 8379578);