#8455684

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=h,e^2=hi,f^2=gh,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, 8362880);