#7768028

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=h,f^2=h,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, 7675224);