#7633269

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

7918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,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, 7540465);