#8458092

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=gh,e^2=ghi,f^2=h,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, 8365288);