#346629

order	512
nilpotency class	3
isoclinism	isoclinic class of #346582
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#555	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253825);