#7712690

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

7260 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)(5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,18)(17,24)(20,25)(21,22)(23,28)(26,27);(1,2,3,4)(13,14,18,16)(15,19,21,22)(17,20,25,24)(23,26,27,28);(6,10)(7,11)(14,16)(15,21)(20,24)(23,27);(5,7,9,11)(6,8,10,12)(13,15,18,21)(14,19,16,22)(17,23,25,27)(20,26,24,28);(5,8)(6,11)(7,10)(9,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26);(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619886);