#7673538

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

7096 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)(15,23)(16,18)(17,20)(19,22)(21,28)(24,25)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,19,17)(14,20,22,23)(16,26,21,24)(18,25,28,27);(2,4)(15,17)(20,23)(24,26)(25,27);(1,2)(3,4)(13,16,19,21)(14,18,22,28)(15,24,17,26)(20,27,23,25);(5,7,8,11)(6,10,9,12)(13,17,19,15)(14,20,22,23)(16,26,21,24)(18,27,28,25);(13,18)(14,21)(15,25)(16,22)(17,27)(19,28)(20,24)(23,26)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580734);