#348279

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#562	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

985 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255475);