#7746730

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

7492 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)(13,14,19,22)(15,18,21,16)(17,25,28,20)(23,24,26,27);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,19,21)(14,16,22,18)(17,26,28,23)(20,27,25,24);(6,10)(8,11)(13,15)(14,16)(17,23)(18,22)(19,21)(20,24)(25,27)(26,28);(13,16,19,18)(14,15,22,21)(17,24,28,27)(20,23,25,26);(5,7)(6,11)(8,10)(9,12)(13,17)(14,20)(15,23)(16,24)(18,27)(19,28)(21,26)(22,25);(5,8,9,11)(6,7,10,12)(13,18,19,16)(14,21,22,15)(17,27,28,24)(20,26,25,23)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=i,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, 7653926);