#120171

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#380
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

586 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^c=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, 27367);