#7652645

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

6926 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);(1,2,3,4)(6,9)(10,12)(13,14,18,19)(15,17,21,26)(16,20,25,27)(22,24,23,28);(5,6,8,9)(7,10,11,12)(14,19)(17,26)(20,27)(24,28);(5,7)(6,10)(8,11)(9,12)(13,15,18,21)(14,17,19,26)(16,23,25,22)(20,28,27,24);(13,16,18,25)(14,20,19,27)(15,22,21,23)(17,24,26,28);(13,17)(14,21)(15,19)(16,24)(18,26)(20,23)(22,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7559841);