#7749866

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7657062);