#8928746

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	#53504	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8835942);