#10368619

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	#53424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10275815);