#7732468

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(5,6,10,9)(7,12,8,11)(13,14)(15,23)(16,17)(18,20)(19,22)(21,28)(24,25)(26,27);(1,2,3,4)(5,7,10,8)(6,11,9,12);(7,8)(11,12)(13,15,19,18)(14,20,22,23)(16,26,21,24)(17,25,28,27);(13,16,19,21)(14,17,22,28)(15,24,18,26)(20,27,23,25);(5,8,10,7)(6,11,9,12)(13,17)(14,21)(15,25)(16,22)(18,27)(19,28)(20,24)(23,26);(5,9,10,6)(7,11,8,12)(13,18,19,15)(14,20,22,23)(16,26,21,24)(17,27,28,25)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7639664);