#7902787

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	#34990	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4521 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7809983);