#7947210

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	#38014	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7854406);