#7642937

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

7382 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)(6,9)(7,11)(13,14)(15,22)(16,17)(18,19)(20,25)(21,26)(23,28)(24,27);(1,2)(3,4)(6,9)(10,12);(5,6)(7,10)(8,9)(11,12)(14,19)(16,21)(20,22)(23,27);(5,7,8,11)(6,10,9,12)(13,15,18,25)(14,20,19,22)(16,23,21,27)(17,24,26,28);(13,16,18,21)(14,17,19,26)(15,23,25,27)(20,24,22,28);(13,17)(14,21)(15,24)(16,19)(18,26)(20,27)(22,23)(25,28)

Polycyclic presentation:

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

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, 7550133);