#283919

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

821 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 191115);