#7684580

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

7049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=i,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, 7591776);