#7870497

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4359 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)(9,10,14,17)(11,15,19,13)(12,20,22,16)(18,21,24,23)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,39,37);(1,2)(3,7)(4,8)(5,6)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(9,11,14,19)(10,13,17,15)(12,18,22,24)(16,21,20,23)(25,27,30,35)(26,31,33,28)(29,34,38,39)(32,37,40,36);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,21)(22,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,16,17,20)(11,18,19,24)(13,21,15,23)(25,28)(26,27)(29,37)(30,31)(32,39)(33,35)(34,40)(36,38);(1,4,5,3)(2,8,6,7)(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,20,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,37)(33,40)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7777693);