#7853278

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

7117 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)(5,6)(7,10)(8,11)(9,12)(13,14)(15,21)(16,24)(17,23)(18,22)(19,25)(20,26)(27,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,17)(18,27)(19,26)(20,25)(22,28)(23,24);(5,6)(7,10)(8,11)(9,12)(13,16,20,18)(14,22,26,24)(15,27,25,17)(19,28,21,23);(13,17,20,27)(14,23,26,28)(15,16,25,18)(19,22,21,24);(5,6)(7,9)(8,11)(10,12)(13,18,20,16)(14,24,26,22)(15,27,25,17)(19,28,21,23);(5,7)(6,10)(8,12)(9,11)(13,19)(14,25)(15,26)(16,28)(17,22)(18,23)(20,21)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7760474);