#351672

order	512
nilpotency class	3
isoclinism	isoclinic class of #347638
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#1667	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

890 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,25)(17,19)(18,23)(20,22)(24,27)(26,28);(13,15,22,28,17,24,16,23)(14,18,25,27,19,26,20,21);(1,2,3,4)(6,9)(10,11)(14,19)(18,26)(20,25)(21,27);(6,10)(8,12)(9,11)

Polycyclic presentation:

2^9|b^2=eh,c^2=fhi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,g^d=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258868);