#7745329

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	#40661	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4534 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7652525);