#7749849

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	#44226	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4336 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,36)(28,39)(29,30)(31,33)(32,37)(34,38)(35,40);(1,2)(3,7)(4,8)(5,6)(25,27)(26,32)(28,29)(30,34)(31,37)(33,36)(35,39)(38,40);(10,11)(15,17)(16,18)(22,23)(26,33)(27,37)(28,38)(30,35);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,31,38)(26,34,33,39)(27,29,37,40)(30,36,35,32);(1,4,5,3)(2,8,6,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29)(26,35)(27,38)(28,37)(30,33)(31,40)(32,34)(36,39);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30,31,35)(26,29,33,40)(27,34,37,39)(28,36,38,32)

Polycyclic presentation:

2^9|a^2=hi,d^2=ghi,e^2=gh,f^2=h,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, 7657045);