#7888127

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 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)(11,12,19,15)(13,16,22,24)(18,20,23,21)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,23)(22,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,33)(34,35)(39,40);(9,11,14,19)(10,15,17,12)(13,18,22,23)(16,21,24,20)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28,30,37)(26,32,34,40)(27,36,38,29)(31,33,35,39);(1,4,5,3)(2,8,6,7)(9,12)(10,11)(13,21)(14,15)(16,23)(17,19)(18,24)(20,22)(25,26)(27,31)(28,32)(29,39)(30,34)(33,36)(35,38)(37,40);(9,13,14,22)(10,16,17,24)(11,18,19,23)(12,20,15,21)(25,29,30,36)(26,33,34,39)(27,37,38,28)(31,32,35,40)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gh,e^2=gi,f^2=i,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, 7795323);