#348197

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#1600	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=eh,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255393);