#7805060

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	#37456	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4542 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)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,26)(27,29)(28,33)(30,34)(31,36)(32,39)(35,38)(37,40);(1,2)(3,7)(4,8)(5,6)(10,11)(12,19)(13,21)(22,23)(25,27,30,36)(26,31,34,29)(28,37,39,35)(32,40,33,38);(1,3,5,4)(2,7,6,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,28,30,39)(26,32,34,33)(27,35,36,37)(29,40,31,38);(1,4,5,3)(2,8,6,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,29,30,31)(26,27,34,36)(28,38,39,40)(32,37,33,35);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27)(26,31)(28,35)(29,34)(30,36)(32,38)(33,40)(37,39);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,33)(27,35)(29,38)(30,39)(31,40)(32,34)(36,37)

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,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, 7712256);