#8932358

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	#47812	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=g,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, 8839554);