#7712807

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	#37455	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4588 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)(9,10,14,11)(12,17,19,15)(13,18,21,16)(20,22,24,23)(29,37)(34,38)(35,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,4,7)(2,6,5,8)(10,11)(12,19)(16,18)(20,24)(26,31)(28,32)(34,38)(36,39);(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27,30,33)(26,32,31,28)(29,35,37,40)(34,39,38,36);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,37)(26,34,31,38)(27,35,33,40)(28,36,32,39)

Polycyclic presentation:

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

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, 7620003);