#7665649

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
prime radical	C₂×#3482	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6858 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572845);