#10325800

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3815 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=g,e^2=gh,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, 10232996);