#7694173

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

7020 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)(6,10)(11,12)(13,14)(15,18)(16,22)(17,25)(19,21)(20,24)(23,26)(27,28);(5,6,9,10)(7,11,8,12)(13,15,20,19)(14,21,24,18)(16,25,27,23)(17,22,26,28);(2,4)(5,7,9,8)(6,11,10,12)(13,16,20,27)(14,22,24,28)(15,23,19,25)(17,18,26,21);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,17,20,26)(14,23,24,25)(15,22,19,28)(16,18,27,21);(5,8,9,7)(6,12,10,11)(13,18)(14,15)(16,26)(17,27)(19,24)(20,21)(22,23)(25,28);(13,19,20,15)(14,21,24,18)(16,25,27,23)(17,28,26,22)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7601369);