#7631261

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

8400 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)(5,6,8,10)(7,9,11,12)(13,14,19,16)(15,17,21,20)(18,24,26,22)(23,27,28,25);(5,6)(7,9)(8,10)(11,12)(13,15,19,21)(14,20,16,17)(18,23,26,28)(22,27,24,25);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,16)(15,21)(22,24)(23,28);(5,7,8,11)(6,9,10,12)(13,16,19,14)(15,17,21,20)(18,24,26,22)(23,25,28,27);(13,17)(14,21)(15,16)(18,25)(19,20)(22,28)(23,24)(26,27);(13,18,19,26)(14,22,16,24)(15,23,21,28)(17,25,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538457);