#8927143

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	#45576	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,32)(28,31)(33,34,39,41)(35,40,36,44)(37,38,47,48)(42,46,45,43);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(18,23)(20,25)(24,26)(27,31)(33,35,39,36)(34,40,41,44)(37,45,47,42)(38,43,48,46);(3,11)(4,12)(6,16)(8,13)(17,18,22,23)(19,24,29,26)(20,21,25,30)(27,28,31,32)(33,36,39,35)(34,40,41,44)(37,45,47,42)(38,46,48,43);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,19,22,29)(18,24,23,26)(20,27,25,31)(21,28,30,32);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,20,22,25)(18,21,23,30)(19,27,29,31)(24,28,26,32)(33,37)(34,38)(35,42)(36,45)(39,47)(40,46)(41,48)(43,44);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,21)(18,25)(19,28)(20,23)(22,30)(24,31)(26,27)(29,32)(33,38)(34,37)(35,43)(36,46)(39,48)(40,45)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^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, 8834339);