#7655284

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

7358 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)(13,14,18,22)(15,17,21,19)(16,24,26,20)(23,27,28,25);(1,2)(3,4)(5,6,10,9)(7,12,8,11);(5,7,10,8)(6,11,9,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,27)(24,25)(26,28);(13,16,18,26)(14,20,22,24)(15,23,21,28)(17,25,19,27);(5,8,10,7)(6,11,9,12)(13,15)(14,17)(16,23)(18,21)(19,22)(20,25)(24,27)(26,28);(5,9,10,6)(7,12,8,11)(13,17)(14,21)(15,22)(16,25)(18,19)(20,28)(23,24)(26,27)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7562480);