#7645387

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4346 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7552583);