#348333

order	512
nilpotency class	3
isoclinism	isoclinic class of #348318
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255529);