#8464264

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	#42893	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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)(3,7)(4,14)(5,10)(6,11)(8,12)(9,13)(15,16)(17,18)(19,24)(20,25)(21,31)(22,27)(23,28)(26,30)(29,32)(33,34,39,42)(35,38,41,36)(37,45,46,40)(43,44,48,47);(1,3,8,6)(2,7,12,11)(4,15,13,5)(9,10,14,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,31)(33,35,39,41)(34,36,42,38)(37,43,46,48)(40,44,45,47);(1,4)(2,9)(3,5)(6,15)(7,16)(8,13)(10,11)(12,14)(17,19,23,32)(18,24,28,29)(20,30,27,21)(22,31,25,26)(33,36,39,38)(34,35,42,41)(37,44,46,47)(40,43,45,48);(1,5,8,15)(2,10,12,16)(3,13,6,4)(7,9,11,14)(17,20)(18,25)(19,30)(21,32)(22,28)(23,27)(24,26)(29,31)(33,37)(34,40)(35,43)(36,44)(38,47)(39,46)(41,48)(42,45);(1,6,8,3)(2,11,12,7)(4,15,13,5)(9,10,14,16)(17,21,23,30)(18,26,28,31)(19,20,32,27)(22,29,25,24)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,7,8,11)(2,3,12,6)(4,16,13,10)(5,14,15,9)(17,22,23,25)(18,27,28,20)(19,31,32,26)(21,29,30,24)(33,38,39,36)(34,41,42,35)(37,44,46,47)(40,43,45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=h,e^2=i,f^2=gi,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, 8371460);