#7927213

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	#38023	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4538 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gi,e^2=h,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, 7834409);