#7749863

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,32,34)(27,35,31,30)(28,33,29,39)(36,40,37,38);(1,2,5,6)(3,7,4,8)(25,27,32,31)(26,30,34,35)(28,37,29,36)(33,40,39,38);(1,3,5,4)(2,7,6,8)(10,11)(15,17)(16,18)(22,23)(25,28,32,29)(26,33,34,39)(27,36,31,37)(30,38,35,40);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,29,32,28)(26,33,34,39)(27,36,31,37)(30,40,35,38);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,30)(26,27)(28,38)(29,40)(31,34)(32,35)(33,36)(37,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,31,32,27)(26,30,34,35)(28,36,29,37)(33,40,39,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=hi,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, 7657059);