#115718

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#386
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1068 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fh,c^b=cf,e^b=eh

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, 22914);