#8927855

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3860 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,23,24)(19,27,28,25)(20,31,32,21)(22,29,30,26)(33,34,39,37)(35,40,36,43)(38,45,46,41)(42,48,44,47);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(18,24)(19,28)(21,31)(26,29)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(3,11)(4,12)(6,16)(8,13)(17,19,23,28)(18,25,24,27)(20,29,32,26)(21,30,31,22)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,20)(18,21)(19,29)(22,27)(23,32)(24,31)(25,30)(26,28);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21)(18,20)(19,30)(22,28)(23,31)(24,32)(25,29)(26,27)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,22,23,30)(18,26,24,29)(19,21,28,31)(20,27,32,25)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,e^2=gh,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, 8835051);