#7687054

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

7048 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)(7,9)(8,10)(11,12);(1,2)(3,4)(13,14)(15,25)(16,21)(17,22)(18,23)(19,24)(20,28)(26,27);(5,6,8,10)(7,9,11,12)(13,15)(14,20)(16,26)(17,23)(18,27)(19,28)(21,22)(24,25);(5,6)(7,9)(8,10)(11,12)(13,16,19,23)(14,21,24,18)(15,26,28,17)(20,22,25,27);(5,7,8,11)(6,9,10,12)(13,17,19,26)(14,22,24,27)(15,23,28,16)(18,20,21,25);(13,18,19,21)(14,23,24,16)(15,27,28,22)(17,25,26,20)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=hi,f^2=i,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, 7594250);