#8932766

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^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, 8839962);