#7654197

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7561393);