#149372

order	512
nilpotency class	4
isoclinism	isoclinic class of #129554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=deh,d^2=hi,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,e^b=eh,g^b=ghi,h^b=hi,d^c=dh,e^c=eh,g^c=ghi,h^c=hi

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, 56568);