#7799535

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

4276 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)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(1,2,4,5)(3,6,7,8)(9,11,15,14)(10,16,19,20)(12,21,18,23)(13,24,17,22)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(11,14)(16,20)(21,23)(22,24)(25,27,31,29)(26,32,28,33)(30,37,38,35)(34,40,36,39);(9,12)(10,17)(11,21)(13,19)(14,23)(15,18)(16,22)(20,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,3,4,7)(2,6,5,8)(9,13,15,17)(10,18,19,12)(11,22,14,24)(16,21,20,23)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(9,14,15,11)(10,16,19,20)(12,23,18,21)(13,24,17,22)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7706731);