#123534

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₂⁴	C₂×C₄²
Frattini, prime radical, cosocle	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂ ▷ C₁

3662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,g^b=gi

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, 30730);