#7768073

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	#38654	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4329 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=h,e^2=g,f^2=gh,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, 7675269);