#7846832

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	#59825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4240 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)(9,10,15,13)(11,16,14,18)(12,17,21,22)(19,23,20,24)(25,26)(27,35)(28,38)(29,34)(30,32)(31,37)(33,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,18)(14,15)(17,23)(20,21)(22,24)(25,27,30,37)(26,31,32,35)(28,36,39,29)(33,40,38,34);(11,14)(16,18)(19,20)(23,24)(26,32)(31,35)(33,38)(34,40);(9,12)(10,17)(11,19)(13,22)(14,20)(15,21)(16,23)(18,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,40)(38,39);(9,13,15,10)(11,18,14,16)(12,17,21,22)(19,23,20,24)(25,28)(26,33)(27,36)(29,37)(30,39)(31,40)(32,38)(34,35);(1,3,4,7)(2,6,5,8)(9,14,15,11)(10,18,13,16)(12,20,21,19)(17,24,22,23)(25,29,30,36)(26,34,32,40)(27,28,37,39)(31,33,35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=i,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, 7754028);