#7633279

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

8306 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)(15,20)(16,22)(17,18)(19,23)(21,25)(24,27)(26,28);(1,2)(3,4)(13,15,19,26)(14,20,23,28)(16,17,25,24)(18,21,27,22);(5,6,8,9)(7,10,11,12)(13,16)(14,21)(15,24)(17,26)(18,20)(19,25)(22,23)(27,28);(13,16)(14,22)(15,17)(18,20)(19,25)(21,23)(24,26)(27,28);(13,17)(14,18)(15,25)(16,26)(19,24)(20,21)(22,28)(23,27);(5,7,8,11)(6,10,9,12)(13,18)(14,17)(15,22)(16,20)(19,27)(21,26)(23,24)(25,28)

Polycyclic presentation:

2^9|a^2=g,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, 7540475);