#7712347

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	#37513	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4478 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gh,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, 7619543);