#7918709

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	#45468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4358 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825905);