#282710

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
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×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

801 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189906);