#8932754

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	#35420	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839950);