#7744730

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

7794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

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, 7651926);