#7733527

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

7276 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)(13,14,19,22)(15,17,20,16)(18,21,27,28)(23,25,26,24);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,19,20)(14,16,22,17)(18,26,27,23)(21,25,28,24);(6,9)(10,11)(13,16,19,17)(14,20,22,15)(18,24,27,25)(21,26,28,23);(5,6)(7,10)(8,9)(11,12)(13,17,19,16)(14,20,22,15)(18,25,27,24)(21,26,28,23);(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,20)(18,21)(19,22)(23,24)(25,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,18)(14,21)(15,23)(16,24)(17,25)(19,27)(20,26)(22,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7640723);