#8097850

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3892 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

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, 8005046);