#7652618

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

8416 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)(5,6,8,9)(7,11,12,10)(13,14,19,21)(15,20,16,25)(17,18,22,28)(23,24,26,27);(1,2)(3,4)(6,9)(10,11)(13,15,19,16)(14,20,21,25)(17,23,22,26)(18,24,28,27);(5,6)(7,10)(8,9)(11,12)(14,21)(18,28)(20,25)(24,27);(5,7)(6,10)(8,12)(9,11)(13,16,19,15)(14,20,21,25)(17,23,22,26)(18,27,28,24);(13,17,19,22)(14,18,21,28)(15,23,16,26)(20,24,25,27);(13,18)(14,22)(15,24)(16,27)(17,21)(19,28)(20,26)(23,25)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,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, 7559814);