#7654263

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	#35031	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4583 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=gh,f^2=h,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, 7561459);