#123884

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#121
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 31080);