#7684866

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

7373 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)(6,9)(10,12)(13,14)(15,25)(16,22)(17,23)(18,24)(19,21)(20,26)(27,28);(5,6,8,9)(7,10,11,12)(13,15,19,26)(14,20,21,25)(16,18,23,27)(17,24,22,28);(2,4)(14,21)(15,26)(16,23)(24,28);(1,2)(3,4)(13,16,19,23)(14,22,21,17)(15,18,26,27)(20,28,25,24);(5,7,8,11)(6,10,9,12)(13,17,19,22)(14,23,21,16)(15,24,26,28)(18,20,27,25);(13,18)(14,24)(15,23)(16,26)(17,25)(19,27)(20,22)(21,28)

Polycyclic presentation:

2^9|b^2=g,d^2=i,e^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, 7592062);