#7655524

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

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

8427 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)(6,9)(7,11);(1,2)(3,4)(6,9)(10,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,19,22,23)(16,17,24,27)(20,26,28,21);(5,7,8,11)(6,10,9,12)(13,16,18,24)(14,20,22,28)(15,17,25,27)(19,26,23,21);(13,17)(14,21)(15,24)(16,25)(18,27)(19,20)(22,26)(23,28);(13,14)(15,19)(16,20)(17,26)(18,22)(21,27)(23,25)(24,28)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^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, 7562720);