#7689587

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

7533 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,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, 7596783);