#7687044

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

7252 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)(5,6,9,12)(7,10,8,11)(13,14,20,19)(15,24,26,21)(16,27,18,22)(17,23,25,28);(5,6)(7,10)(8,11)(9,12)(13,15)(14,21)(16,17)(18,25)(19,24)(20,26)(22,23)(27,28);(1,2,3,4)(13,16,20,18)(14,22,19,27)(15,25,26,17)(21,28,24,23);(1,2)(3,4)(13,17,20,25)(14,23,19,28)(15,16,26,18)(21,22,24,27);(5,7,9,8)(6,10,12,11)(13,18,20,16)(14,22,19,27)(15,25,26,17)(21,23,24,28);(5,8,9,7)(6,11,12,10)(13,19,20,14)(15,24,26,21)(16,27,18,22)(17,28,25,23)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7594240);