#7709822

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	#45784	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617018);