#8436609

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	#52370	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,e^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, 8343805);