#7849041

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

4153 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7756237);