#7772007

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	#44337	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4226 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)(25,26,30,29)(27,31,35,32)(28,36,38,33)(34,40,37,39);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,27)(26,31)(28,34)(29,32)(30,35)(33,39)(36,40)(37,38);(1,3,5,4)(2,7,6,8)(10,15)(13,22)(16,23)(20,24)(25,27,30,35)(26,32,29,31)(28,34,38,37)(33,40,36,39);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,27)(26,31)(28,37)(29,32)(30,35)(33,40)(34,38)(36,39);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,28)(26,33)(27,34)(29,36)(30,38)(31,39)(32,40)(35,37);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,29,30,26)(27,32,35,31)(28,36,38,33)(34,40,37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7679203);