#7633309

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

8566 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,f^2=hi,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540505);