#8041188

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	#49554	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7948384);