#7685073

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

7424 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)(6,10)(11,12);(5,6,9,10)(7,11,8,12)(13,14)(15,20)(16,19)(17,21)(18,23)(22,26)(24,27)(25,28);(1,2,3,4)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(1,2)(3,4)(13,16,17,24)(14,19,21,27)(15,22,23,28)(18,25,20,26);(5,7,9,8)(6,11,10,12)(13,15)(14,20)(16,22)(17,23)(18,21)(19,26)(24,28)(25,27);(5,8,9,7)(6,12,10,11)(13,14)(15,18)(16,19)(17,21)(20,23)(22,25)(24,27)(26,28)

Polycyclic presentation:

2^9|b^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, 7592269);