#8445957

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=gh,e^2=gi,f^2=g,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, 8353153);