#7813022

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	#47876	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,e^2=h,f^2=g,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, 7720218);