#8932215

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

3773 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)(3,5)(4,13)(6,9)(7,8)(10,12)(11,15)(14,16)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(35,38)(40,42)(43,45)(47,48);(1,3)(2,5)(4,14)(6,11)(7,12)(8,10)(9,15)(13,16)(17,18,22,24)(19,25,27,20)(21,23,30,29)(26,31,32,28)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(1,4,8,6)(2,9,7,13)(3,11,10,14)(5,16,12,15)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,5,8,12)(2,10,7,3)(4,15,6,16)(9,11,13,14)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,6,8,4)(2,9,7,13)(3,11,10,14)(5,15,12,16)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,7,8,2)(3,12,10,5)(4,9,6,13)(11,16,14,15)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=gh,e^2=gi,f^2=h,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, 8839411);