#7805062

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4448 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)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(1,3,5,8)(2,6,4,7)(9,10,14,18)(11,16,12,15)(13,21,22,17)(19,23,20,24)(26,28)(27,34)(29,37)(38,39);(1,2,5,4)(3,6,8,7)(9,11,14,12)(10,15,18,16)(13,20,22,19)(17,24,21,23)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,4,5,2)(3,7,8,6)(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,27)(26,33)(28,31)(29,35)(30,34)(32,39)(36,38)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7712258);