#7630851

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

7322 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)(5,6,10,9)(7,12,11,8)(13,14,20,18)(15,23,22,19)(16,25,17,21)(24,27,26,28);(5,7)(6,8)(9,12)(10,11)(13,15)(14,19)(16,26)(17,24)(18,23)(20,22)(21,28)(25,27);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,16,20,17)(14,21,18,25)(15,24,22,26)(19,27,23,28);(13,17,20,16)(14,21,18,25)(15,24,22,26)(19,28,23,27);(13,18,20,14)(15,23,22,19)(16,25,17,21)(24,28,26,27);(5,9,10,6)(7,12,11,8)(13,19,20,23)(14,22,18,15)(16,27,17,28)(21,26,25,24)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538047);