#8901076

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8808272);